Michael Hothorn

Youssef Belkhadir

Marlene Dreux

Tsegaye Dabi

Joseph P Noel

Ian A Wilson

Joanne Chory
email: chory@salk.edu.

Structural basis of steroid hormone perception by the receptor kinase BRI1

The LRR-RK BRI1 [START_REF] Li | A putative leucine-rich repeat receptor kinase involved in brassinosteroid signal transduction[END_REF][START_REF] Belkhadir | Brassinosteroid signaling: a paradigm for steroid hormone signaling from the cell surface[END_REF] controls a steroid signalling pathway essential for plant growth [START_REF] Vert | Molecular mechanisms of steroid hormone signaling in plants[END_REF] . While animal steroid receptors are found predominantly in the nucleus [START_REF] Mangelsdorf | The nuclear receptor superfamily: the second decade[END_REF] , BRI1 is localised at the plasma-membrane and in endosomes [START_REF] Geldner | Endosomal signaling of plant steroid receptor kinase BRI1[END_REF] . The following model for BRI1 activation has been proposed: In the absence of brassinosteroid, BRI1's kinase domain is kept in a basal state by its auto-inhibitory C-terminal tail [START_REF] Wang | Autoregulation and homodimerization are involved in the activation of the plant steroid receptor BRI1[END_REF] , as well as by interaction with the inhibitor protein BKI1 [START_REF] Wang | Brassinosteroids regulate dissociation of BKI1, a negative regulator of BRI1 signaling, from the plasma membrane[END_REF] . Hormone binding to the extracellular domain of BRI1 [START_REF] Wang | BRI1 is a critical component of a plasmamembrane receptor for plant steroids[END_REF][START_REF] He | Perception of brassinosteroids by the extracellular domain of the receptor kinase BRI1[END_REF] in a region that includes a ~70 amino acid 'island' domain [START_REF] Kinoshita | Binding of brassinosteroids to the extracellular domain of plant receptor kinase BRI1[END_REF] , causes a change in the receptor (a conformational change in a preformed homodimer [START_REF] Wang | Autoregulation and homodimerization are involved in the activation of the plant steroid receptor BRI1[END_REF] or receptor dimerisation), leading to autophosphorylation of the BRI1 kinase domain [START_REF] Wang | Identification and functional analysis of in vivo phosphorylation sites of the Arabidopsis BRASSINOSTEROID-INSENSITIVE1 receptor kinase[END_REF] , release of its C-terminal tail [START_REF] Wang | Autoregulation and homodimerization are involved in the activation of the plant steroid receptor BRI1[END_REF] , and transphosphorylation of the inhibitor BKI1 [START_REF] Wang | Brassinosteroids regulate dissociation of BKI1, a negative regulator of BRI1 signaling, from the plasma membrane[END_REF][START_REF] Jaillais | Tyrosine phosphorylation controls brassinosteroid receptor activation by triggering membrane release of its kinase inhibitor[END_REF] . BKI1 then dissociates from the membrane, allowing BRI1 to interact with a family of smaller LRR-RKs [START_REF] Chinchilla | One for all: the receptor-associated kinase BAK1[END_REF] , including the BRI1 ASSOCIATED KINASE 1 (BAK1) [START_REF] Nam | BRI1/BAK1, a receptor kinase pair mediating brassinosteroid signaling[END_REF][START_REF] Li | BAK1, an Arabidopsis LRR receptor-like protein kinase, interacts with BRI1 and modulates brassinosteroid signaling[END_REF] . The kinase domains of BRI1 and BAK1 transphosphorylate each other on multiple sites [START_REF] Wang | Sequential transphosphorylation of the BRI1/BAK1 receptor kinase complex impacts early events in brassinosteroid signaling[END_REF] , and the fully activated receptor triggers downstream signalling events [START_REF] Kim | Brassinosteroid signal transduction from receptor kinases to transcription factors[END_REF] , resulting in major changes in nuclear gene expression [START_REF] Vert | Molecular mechanisms of steroid hormone signaling in plants[END_REF] . BRI1's architecture is reminiscent of animal Toll-like innate immunity receptors (TLRs), and notably several plant LRR-RKs are immunity receptors [START_REF] Gómez-Gómez | FLS2: an LRR receptor-like kinase involved in the perception of the bacterial elicitor flagellin in Arabidopsis[END_REF] . It was, thus, reasonable to assume that the BRI1 ectodomain would form a TLR-like horseshoe structure [START_REF] Choe | Crystal structure of human toll-like receptor 3 (TLR3) ectodomain[END_REF] that binds its ligand along a dimer interface, as observed in several TLRs [START_REF] Liu | Structural basis of toll-like receptor 3 signaling with double-stranded RNA[END_REF][START_REF] Park | The structural basis of lipopolysaccharide recognition by the TLR4-MD-2 complex[END_REF] . Here, we report the structure of the ligand binding domain of Arabidopsis BRI1 in its free form and bound to the steroid brassinolide, and show that BRI1 folds into a superhelical assembly, whose interior provides the hormone-binding site. Comparison of the free and hormone-bound structures, combined with genetic data, suggests a novel activation mechanism for BRI1 that is distinct from TLRs.

Overall structure of the BRI1 ectodomain

BRI1 was expressed in baculovirus-infected insect cells and the secreted ectodomain was purified by tandem-affinity and size-exclusion chromatography. The crystal structure was solved to 2.5 Å resolution by single isomorphous replacement (see Methods, Supplementary Table 1 and Fig. 1). BRI1 does not adopt the anticipated TLR-horseshoe structure but forms a right-handed superhelix composed of 25 LRRs (Fig. 1a). The helix completes one full turn, with a rise of ~70 Å. The concave surface, that determines the curvature of the solenoid [START_REF] Bella | The leucine-rich repeat structure[END_REF] , is formed by αand 3 10 helices (green in Fig. 1a) that cause inner and outer diameters of ~30 and ~60 Å, respectively. The overall curvature of BRI is similar to TLR3 [START_REF] Choe | Crystal structure of human toll-like receptor 3 (TLR3) ectodomain[END_REF] (Fig. 1b), but, while the TLR3 ectodomain is essentially flat, BRI1 is highly twisted (Fig. 1b). Such twisted assemblies of LRRs have been observed previously with bacterial effector [START_REF] Evdokimov | Unusual molecular architecture of the Yersinia pestis cytotoxin YopM: a leucine-rich repeat protein with the shortest repeating unit[END_REF] and adhesion proteins [START_REF] Schubert | Structure of internalin, a major invasion protein of Listeria monocytogenes, in complex with its human receptor E-cadherin[END_REF] , and with the plant defence protein PGIP 30 (Supplementary Fig. 2). The twist of PGIP's LRR domain is caused by a non-canonical, second β-sheet that is oriented perpendicular to the central β-sheet forming the inner surface of the solenoid [START_REF] Matteo | The crystal structure of polygalacturonase-inhibiting protein (PGIP), a leucinerich repeat protein involved in plant defense[END_REF] . Additional β-sheets are also present in our structure (blue in Fig. 1a, Supplementary Fig. 3), but in the case of the much larger BRI1 ectodomain result in a superhelical assembly (Fig. 1a). The second β-strand in PGIP and in BRI1 is followed by an Ile-Pro spine that runs along the outer surface of the helix and provides packing interactions between consecutive LRRs (Fig. 2a and ref. [START_REF] Matteo | The crystal structure of polygalacturonase-inhibiting protein (PGIP), a leucinerich repeat protein involved in plant defense[END_REF] ). Both structural features are directly linked to the Lt/sGxIP consensus sequence of the plant-specific LRR subfamily [START_REF] Kajava | Structural diversity of leucine-rich repeat proteins[END_REF] (Fig. 2c, Supplementary Fig. 4 and Table 2). Because this consensus sequence is found in other plant RKs, these receptors may also harbour twisted LRR domains (Fig. 2c), making BRI1 the primary template for the study of diverse signalling pathways in plants [START_REF] Clark | The CLAVATA1 gene encodes a putative receptor kinase that controls shoot and floral meristem size in Arabidopsis[END_REF][START_REF] Nadeau | Control of stomatal distribution on the Arabidopsis leaf surface[END_REF][START_REF] Gómez-Gómez | FLS2: an LRR receptor-like kinase involved in the perception of the bacterial elicitor flagellin in Arabidopsis[END_REF] .

N-and C-terminal flanking regions that cap the hydrophobic core of the BRI1 solenoid are similar to caps previously described for PGIP [START_REF] Matteo | The crystal structure of polygalacturonase-inhibiting protein (PGIP), a leucinerich repeat protein involved in plant defense[END_REF] (Supplementary Figure. 5). Notably, not are only these caps stabilised by disulfide bridges, but 5 additional disulfide bonds link consecutive LRR segments in the N-terminal half of the BRI1 ectodomain (Fig. 2b, Supplementary Fig. 4 and Table 2).

The island domain

The island domain in BRI1 corresponds to a large insertion in the regular repeat-structure between LRRs 21 and 22 (residues 584-654) (Fig. 1a). The resulting ~70 residue segment forms a small domain that folds back into the interior of the superhelix, where it makes extensive polar and hydrophobic interactions with LRRs 13-25 (Fig 1a, Supplementary Fig. 6 and Table 2). The domain fold is characterised by an anti-parallel β-sheet, which is sandwiched between the LRR core and a 3 10 helix and stabilised by a disulfide bridge (Fig. 3a, Supplementary Fig. 4). The loss-of-function alleles bri1-9 (Ser662Phe, weak) [START_REF] Noguchi | Brassinosteroid-insensitive dwarf mutants of Arabidopsis accumulate brassinosteroids[END_REF] and bri1-113 (Gly611Glu, strong) 2 map to this island domain -LRR interface (Supplementary Fig. 6), and likely interfere with folding of the island domain [START_REF] Jin | Allele-specific suppression of a defective brassinosteroid receptor reveals a physiological role of UGGT in ER quality control[END_REF] . Two long loops that connect the island domain to the LRR core appear partially disordered in the unliganded receptor (Supplementary Fig. 7). The insertion of a folded domain into the LRR repeat has not been observed in other LRR receptor structures and is likely an adaptation to the challenge of sensing a small steroid ligand, compared, for example, to recognising larger ligands, such as proteins, nucleic acids, or lipids [START_REF] Liu | Structural basis of toll-like receptor 3 signaling with double-stranded RNA[END_REF][START_REF] Park | The structural basis of lipopolysaccharide recognition by the TLR4-MD-2 complex[END_REF] .

We next solved a 2.5 Å co-crystal structure with brassinolide, a potent Arabidopsis steroid that binds BRI1 with nanomolar affinity [START_REF] Wang | BRI1 is a critical component of a plasmamembrane receptor for plant steroids[END_REF][START_REF] Fujioka | The Arabidopsis deetiolated2 mutant is blocked early in brassinosteroid biosynthesis[END_REF] . One molecule of brassinolide per BRI1 monomer binds in close proximity to the island domain (Fig. 3 a-c), which was previously implicated in steroid binding [START_REF] Wang | BRI1 is a critical component of a plasmamembrane receptor for plant steroids[END_REF][START_REF] Kinoshita | Binding of brassinosteroids to the extracellular domain of plant receptor kinase BRI1[END_REF] . Our structure reveals that the LRR superhelix and the island domain both extensively contribute to formation of the hormone binding site. The A-D rings of the steroid bind to a hydrophobic surface which is provided by LRRs 23-25 and that maps to the inner side of the BRI1 superhelix (Fig. 3b,d, Supplementary Fig. 8). The alkyl chain of the hormone fits into a small pocket formed by residues originating from LRRs 21 and 22 (Ile563, Trp564, Met657, Phe658) and from two loops connecting the island domain with the LRR core (Fig. 3d). The hydrophobic nature and restricted size of this pocket now explain why steroid ligands with bulkier or charged alkyl side chains, such as the arthropod steroid ecdysone (Supplementary Fig. 8), cannot be recognised by BRI1 [START_REF] Wang | BRI1 is a critical component of a plasmamembrane receptor for plant steroids[END_REF] . A few polar interactions with brassinolide's second diol moiety (Fig. 3d) are established with Tyr597 and main chain atoms from His645 and Ser647 in the island domain, and are mediated by water molecules (Fig. 3d). Mutation of the neighbouring Gly644 to Asp may interfere with this hydrogen bonding network, and explain why this mutation greatly reduces the binding activity of the receptor [START_REF] Wang | BRI1 is a critical component of a plasmamembrane receptor for plant steroids[END_REF] and causes the loss-of-function phenotype bri1-6 [START_REF] Noguchi | Brassinosteroid-insensitive dwarf mutants of Arabidopsis accumulate brassinosteroids[END_REF] (Fig. 3d). No polar contacts are observed with the seven-membered B-ring lactone (Fig. 3d), consistent with B-ring modifications as found in e.g. castasterone (Supplementary Fig. 8) being tolerated by BRI1 [START_REF] Wang | BRI1 is a critical component of a plasmamembrane receptor for plant steroids[END_REF][START_REF] Back | Structure-Activity Studies of Brassinosteroids and the Search for Novel Analogues and Mimetics with Improved Bioactivity[END_REF] .

The steroid-complex reveals a hormone-binding site that involves a much larger portion of the LRR domain than previously anticipated [START_REF] Kinoshita | Binding of brassinosteroids to the extracellular domain of plant receptor kinase BRI1[END_REF] . Major interactions between the steroid and the BRI1 ectodomain originate from the very C-terminal LRRs 23-25, which brings the hormone in close proximity to the membrane (Fig. 3a,d). Importantly, while there is a significant hormone-receptor interface (550 Å 2 ) for such a small molecule ligand, large parts of the steroid are exposed to the solvent, including the 2α,3α-diol moiety in brassinolide that is important for biological activity [START_REF] Back | Structure-Activity Studies of Brassinosteroids and the Search for Novel Analogues and Mimetics with Improved Bioactivity[END_REF] . Thus, protein-protein interactions may be involved in the recognition of the steroid ligand, with the hormone itself providing a docking platform. Importantly, steroid binding induces a conformational rearrangement and fixing of the island domain, which becomes fully ordered and competent to participate in protein-protein interactions that could be critical for receptor activation (Supplementary Fig. 7).

A protein interaction platform

Four known BRI1 missense alleles map to the inner surface of last 5 LRRs (Fig. 5a). This surface is not masked by carbohydrate (Supplementary Fig. 9) and contains both the hormone-binding site and the island domain (Fig. 3a,d and4a,c). Three mutations cluster in a loop connecting the island domain with LRR 22 (Fig. 5a). This loop is partially disordered in the unliganded structure but is well-defined in the brassinolide complex (Supplementary Fig. 7). We speculate that this loop, when ordered, is engaged in protein-protein interactions that are critical for receptor activation, and that missense alleles in BRI1 modulate these interactions: The gain-of-function allele sud1 36 (Gly643-Glu) may establish contact with Ser623 in the island domain, and lead to an ordered loop even in the absence of steroid ligand (Supplementary Fig. 10). Mutation of the neighbouring Gly644 to Asp causes the loss-of-function phenotype bri1-6 [START_REF] Noguchi | Brassinosteroid-insensitive dwarf mutants of Arabidopsis accumulate brassinosteroids[END_REF] (see above, Fig. 3d,4a), and mutation of conserved Thr649 to Lys inactivates barley BRI1 [START_REF] Gruszka | New allele of HvBRI1 gene encoding brassinosteroid receptor in barley[END_REF] . These mutations, when modelled in silico, induce steric clashes with residues in the island domain and in the underlying LRR domain (Supplementary Fig. 10), and thus may distort the position of the loop. Interestingly, bri1-102, a strong loss-of-function mutation (Thr750-Ile) [START_REF] Friedrichsen | Brassinosteroid-insensitive-1 is a ubiquitously expressed leucine-rich repeat receptor serine/threonine kinase[END_REF] that does not affect steroid binding [START_REF] Wang | BRI1 is a critical component of a plasmamembrane receptor for plant steroids[END_REF] , maps to a distinct surface area in LRR 25 (Fig. 4a). Thus protein-protein interactions critical for receptor activation may not be restricted to the island domain, but also involve residues from the LRR core.

Receptor activation

BRI1 has been shown to exist at least partially as a omooligomer in planta [START_REF] Wang | Autoregulation and homodimerization are involved in the activation of the plant steroid receptor BRI1[END_REF][START_REF] Hink | Fluorescence fluctuation analysis of Arabidopsis thaliana somatic embryogenesis receptor-like kinase and brassinosteroid insensitive 1 receptor oligomerization[END_REF][START_REF] Russinova | Heterodimerization and endocytosis of Arabidopsis brassinosteroid receptors BRI1 and AtSERK3 (BAK1)[END_REF] . Thus steroid binding to the island domain and the concomitant rearrangements of the island domain loop could induce a conformational change in a preformed BRI1 homodimer [START_REF] Wang | Autoregulation and homodimerization are involved in the activation of the plant steroid receptor BRI1[END_REF] , or allow for ligand-dependent dimerisation of the BRI1 ectodomain. However, models of BRI1 dimers that bring the C-termini of their ectodomains into close proximity (n.b. it is known that the cytoplasmic kinase domains of BRI1 can interact [START_REF] Wang | Autoregulation and homodimerization are involved in the activation of the plant steroid receptor BRI1[END_REF][START_REF] Jaillais | Tyrosine phosphorylation controls brassinosteroid receptor activation by triggering membrane release of its kinase inhibitor[END_REF] ) and that make use of the interaction surface outlined above, encounter steric clashes with the N-terminal LRRs (Supplementary Fig. 11). Furthermore, in contrast to TLR ectodomains, which in crystals tend to form homodimers even in the absence of ligand [START_REF] Choe | Crystal structure of human toll-like receptor 3 (TLR3) ectodomain[END_REF][START_REF] Liu | Structural basis of toll-like receptor 3 signaling with double-stranded RNA[END_REF] , dimers cannot be seen in BRI1 crystals, which grew under the same acidic pH conditions that are typically associated with the plant cell wall. The largest interface area between two neighbouring BRI1 molecules amounts to only ~1.5% of the total accessible surface area, consistent with the high solvent content of our crystals (see Supplementary Methods). The main crystal contact involves a head-to-head arrangement of two BRI1 monomers, a configuration that would place the cytoplasmic kinase domains far apart (Supplementary Fig. 12). The recombinant BRI1 ectodomain eluates as a monomer in the absence of steroid ligand, and shows no tendency to dimer-or oligomerise in the presence of a ~4x molar excess of brassinolide in size-exclusion chromatography experiments (Fig. 4b).

Our analyses suggest that the superhelical BRI1 LRR domain alone has no tendency to oligomerise, indicating that BRI1 receptor activation may not be mediated by ligandinduced homodimerisation of the ectodomain (as described for TLRs [START_REF] Liu | Structural basis of toll-like receptor 3 signaling with double-stranded RNA[END_REF][START_REF] Park | The structural basis of lipopolysaccharide recognition by the TLR4-MD-2 complex[END_REF] ) or by conformational changes in preformed homodimers [START_REF] Wang | Autoregulation and homodimerization are involved in the activation of the plant steroid receptor BRI1[END_REF] . We do not want to dismiss the fact that the cytoplasmic kinase domain of BRI1 can dimerise [START_REF] Jaillais | Tyrosine phosphorylation controls brassinosteroid receptor activation by triggering membrane release of its kinase inhibitor[END_REF] , or that BRI1 homooligomers are present in vivo [START_REF] Wang | Autoregulation and homodimerization are involved in the activation of the plant steroid receptor BRI1[END_REF][START_REF] Hink | Fluorescence fluctuation analysis of Arabidopsis thaliana somatic embryogenesis receptor-like kinase and brassinosteroid insensitive 1 receptor oligomerization[END_REF][START_REF] Russinova | Heterodimerization and endocytosis of Arabidopsis brassinosteroid receptors BRI1 and AtSERK3 (BAK1)[END_REF] . However, our structures reinforce the notion that homooligomerisation of BRI1 may be constitutive and independent of ligand stimulus [START_REF] Hink | Fluorescence fluctuation analysis of Arabidopsis thaliana somatic embryogenesis receptor-like kinase and brassinosteroid insensitive 1 receptor oligomerization[END_REF] . The presence of an interaction platform that undergoes conformational changes upon steroid binding and that harbours several loss-and gain-of-function alleles suggests that interaction with another protein factor may control BRI1 activation.

Discussion

The structure of the BRI1 ectodomain offers several new insights, and its twisted shape will likely characterise the architecture of many plant LRR-RKs 1 . The presence of a folded domain-insert appears to be an adaptation to recognition of a small molecule ligand, a challenge that smaller LRR proteins have met by generating loop insertions into their capping motifs [START_REF] Han | Antigen recognition by variable lymphocyte receptors[END_REF] . This fascinating mode of ligand recognition reveals how steroids can be sensed at membranes and rationalises a large set of genetic and biochemical findings [START_REF] Li | A putative leucine-rich repeat receptor kinase involved in brassinosteroid signal transduction[END_REF][START_REF] Wang | BRI1 is a critical component of a plasmamembrane receptor for plant steroids[END_REF][START_REF] Noguchi | Brassinosteroid-insensitive dwarf mutants of Arabidopsis accumulate brassinosteroids[END_REF][START_REF] Friedrichsen | Brassinosteroid-insensitive-1 is a ubiquitously expressed leucine-rich repeat receptor serine/threonine kinase[END_REF] . Different BRI1 receptor activation mechanisms have been proposed including liganddependent dimerisation as seen for TLRs [START_REF] Liu | Structural basis of toll-like receptor 3 signaling with double-stranded RNA[END_REF][START_REF] Park | The structural basis of lipopolysaccharide recognition by the TLR4-MD-2 complex[END_REF] and ligand-induced conformational changes in preformed homodimers [START_REF] Wang | Autoregulation and homodimerization are involved in the activation of the plant steroid receptor BRI1[END_REF] . Our analyses suggest that the superhelical shape of the BRI1 ectodomain is incompatible with homodimerisation, and that the isolated ectodomain behaves as a monomer even in the presence of steroid. These findings leave us with the alternative hypothesis that another protein factor could bind to the interaction platform in BRI1 that would minimally encompass the steroid ligand, LRRs 21-25 and parts of the island domain (Fig. 4a). While it is possible that an unknown protein fulfils this role and provides a dimerisation interface for two BRI1 molecules (as seen for example for TLR4 [START_REF] Park | The structural basis of lipopolysaccharide recognition by the TLR4-MD-2 complex[END_REF] ), genetic and biochemical screens have not uncovered this protein. It is thus possible that the small receptor kinase BAK1 acts as a direct brassinosteroid co-receptor, as suggested previously [START_REF] Vert | Molecular mechanisms of steroid hormone signaling in plants[END_REF][START_REF] Nam | BRI1/BAK1, a receptor kinase pair mediating brassinosteroid signaling[END_REF] . It has been demonstrated that BAK1 is a genetic component of the brassinosteroid pathway [START_REF] Nam | BRI1/BAK1, a receptor kinase pair mediating brassinosteroid signaling[END_REF][START_REF] Li | BAK1, an Arabidopsis LRR receptor-like protein kinase, interacts with BRI1 and modulates brassinosteroid signaling[END_REF] , that BRI1 and BAK1 interact in a steroid-dependent manner [START_REF] Wang | Sequential transphosphorylation of the BRI1/BAK1 receptor kinase complex impacts early events in brassinosteroid signaling[END_REF] and that both receptors trans-phosphorylate each other upon ligand stimulus [START_REF] Wang | Sequential transphosphorylation of the BRI1/BAK1 receptor kinase complex impacts early events in brassinosteroid signaling[END_REF] . Notably, a homology model of the BAK1 ectodomain (Supplementary Fig. 13) is compatible in size and shape with the interaction platform in BRI1, and the BAK1 elg allele, which maps to the BAK1 ectodomain (Supplementary Fig. 14), renders plants hypersensitive to brassinosteroid treatment [START_REF] Whippo | A brassinosteroid-hypersensitive mutant of BAK1 indicates that a convergence of photomorphogenic and hormonal signaling modulates phototropism[END_REF] . We speculate that the sud1, bri1-6, bri1-102 and elg mutations modulate the interaction between the BRI1 and BAK1 ectodomains in a brassinosteroid-dependent manner (Supplementary Fig. 14). The demonstration that BAK1 is essential for brassinosteroid sensing may have been obscured owing to genetic redundancy [START_REF] Nam | BRI1/BAK1, a receptor kinase pair mediating brassinosteroid signaling[END_REF] , with at least two BAK1-like proteins interacting with BRI1 in vivo [START_REF] He | BAK1 and BKK1 regulate brassinosteroid-dependent growth and brassinosteroidindependent cell-death pathways[END_REF][START_REF] Karlova | The Arabidopsis SOMATIC EMBRYOGENESIS RECEPTOR-LIKE KINASE1 protein complex includes BRASSINOSTEROID-INSENSITIVE1[END_REF] . We recently overcame this limitation by showing that the BRI1 inhibitor protein BKI1 blocks the interaction between the BAK1 and BRI1 kinase domains [START_REF] Jaillais | Tyrosine phosphorylation controls brassinosteroid receptor activation by triggering membrane release of its kinase inhibitor[END_REF] . Importantly, transgenic lines that constitutively deliver BKI1 to the site of BRI1 signalling, resemble strong BRI1 loss-of-function mutants, suggesting an important role for receptor -co-receptor association in brassinosteroid signal initiation [START_REF] Jaillais | Tyrosine phosphorylation controls brassinosteroid receptor activation by triggering membrane release of its kinase inhibitor[END_REF] . Future studies will undoubtedly test this heteromerisation model and dissect the relative contributions of the BRI1 and BAK1 ectodomains, their transmembrane segments and their cytoplasmic kinase domains to receptor activation. It will become important then to understand how BAK1 could serve as co-receptor for other LRR-RK signalling pathways [START_REF] Gómez-Gómez | FLS2: an LRR receptor-like kinase involved in the perception of the bacterial elicitor flagellin in Arabidopsis[END_REF][START_REF] Chinchilla | One for all: the receptor-associated kinase BAK1[END_REF] .

Methods Summary

The StrepII-9xHis fused BRI1 ectodomain (residues 29-788) was produced by secreted expression in baculovirus-infected insect cells, harvested 4 d post-infection by ultrafiltration and purified by tandem affinity chromatography, and by gel filtration. BRI1 was concentrated to 15 mg/ml and crystallised by vapour diffusion using a reservoir solution containing 14% PEG 4,000, 0.2 M (NH 4 ) 2 SO 4 , 0.1 M citric acid (pH 4.0). The brassinolide complex was obtained by co-crystallisation. Diffraction data to 2.5 Å resolution were collected on a rotating anode X-ray generator and at beam-line 8.2.1 of the Advanced Light a, Overview of the C-terminal surface area (in blue) that is not masked by carbohydrate. Brassinolide is shown in yellow, the island domain in orange, and genetic alleles connected in magenta, b, Analytical gel-filtration 280 nm absorbance trace. The free ectodomain eluates as a monomer (black line), as does a putative complex with brassinolide (red line). Void (V 0 ) and total volume (V t ) are shown together with elution volumes for molecular weight standards (A, aldolase, MW 158,000 Da; B, conalbumin, MW 75,000 Da). The estimated molecular weight for the monomer peak is ~125 kDa. The approximate molecular weight of the purified BRI1 is 110 kDa.

Figure 1 .

 1 Figure 1. The BRI1 ectodomain forms a superhelical assembly a, Ribbon diagram of the BRI1 LRR domain (front, back and top). The canonical LRR βsheet is shown in orange, and the additional plant-specific β-sheets in blue. Helices are shown in green, and the island domain is depicted in red. b, Structural comparison of the BRI1 (shown as yellow C α trace) and TLR3 (in blue, pdb-id: lziw)27 ectodomains. The structures superimpose with an r.m.s.d. of 4.2 Å between 341 corresponding C α atoms. Side and top views are shown. The island domain has been omitted for clarity.

Figure 2 .

 2 Figure 2. Plant-specific sequence fingerprints result in a superhelical BRI1 ectodomain a, Ribbon diagram of the convex side of BRI1 LRRs 9-25 (in yellow). The non-canonical βstrands and the Ile-Pro spine are shown in dark and light blue, respectively, b, Top view of the BRI1 ectodomain (in blue) with disulfide bridges shown in yellow. The N-and Cterminal caps are highlighted in pink, c, Sequence alignment of LRRs in BRI1, other plant receptor kinases[START_REF] Nadeau | Control of stomatal distribution on the Arabidopsis leaf surface[END_REF][START_REF] Gómez-Gómez | FLS2: an LRR receptor-like kinase involved in the perception of the bacterial elicitor flagellin in Arabidopsis[END_REF][START_REF] Nishimura | HAR1 mediates systemic regulation of symbiotic organ development[END_REF][START_REF] Wang | BRI1 is a critical component of a plasmamembrane receptor for plant steroids[END_REF][START_REF] Kim | Brassinosteroid signal transduction from receptor kinases to transcription factors[END_REF][START_REF] Choe | Crystal structure of human toll-like receptor 3 (TLR3) ectodomain[END_REF] and PGEP[START_REF] Jin | Allele-specific suppression of a defective brassinosteroid receptor reveals a physiological role of UGGT in ER quality control[END_REF] . The canonical LRR consensus sequences are highlighted in blue, and plant-specific motifs are in green.

Figure 3 .

 3 Figure 3. The steroid hormone binding site maps to C-terminal inner surface of the superhelix a, Brassinolide (in yellow sticks) binds to a surface provided by the LRR domain (in blue) and by parts of the island domain (green ribbon), b, Location of the steroid in centre of the BRI1 superhelix. c, Close-up view of the brassinolide in two orientations, including an omit 2F o -F c electron density map contoured at 1.5 σ. d, Protein-hormone interactions in the BRI1 steroid binding site. Ribbon diagram of LRRs 21-25 (in grey) are shown together with parts of the island domain (in green). Contacting residues are shown in full side-chain representation, polar interactions as dotted lines, and water molecules as red balls, bril-6 (Gly644Asp) is depicted in magenta.

Figure 4 .

 4 Figure 4. N-linked glycans mask large surface areas of the BRI1 superhelix Oligomannose core stmctures (containing two N-acetylglucosamines and three mannose units) as found in insect cells and plants were modelled onto the nine glycosylation sites in the structure to visualise the BRI1 surface that is potentially masked by carbohydrate. The LRR domain in surface representation is shown in blue, and the glycan structures are highlighted in yellow. A ribbon diagram of the island domain (in green) and the steroid ligand (in yellow) is included. The views are a, front b, back and c, perspective along the superhelix from the C-terminus.

Figure 5 .

 5 Figure 5. An accessible membrane-proximal region of BRI1 may provide a protein-protein interaction platform

Nature. Author manuscript; available in PMC 2012 February 15.

Acknowledgments

We thank J. Vanhnasy and W. Yu for maintaining insect cell stocks, M. Jinek and B.W. Han for advice, W. Kwiatowski for excellent maintenance of the Salk X-ray equipment, Y. Jaillais for discussion, and F. V. Chisari for encouragement and support. This work was supported by the Howard Hughes Medical Institute and a grant from the National Science Foundation (IOS-0649389) to J.C. M.H. was supported by long-term fellowships from the European Molecular Biology Organisation and the International Human Frontier Science Program Organisation. Y.B. was a Howard Hughes Medical Institute fellow of the Life Sciences Research Foundation and was also a recipient of the Philippe Foundation. I.A.W. was supported by NIH grant AI042266 and by the Skaggs Institute for Chemical Biology.

Author Information Atomic coordinates and structure factors for the reported crystal structures have been deposited in the Protein Data Bank under accession numbers 3RIZ for the unliganded BRI1 ectodomain and 3RJ0 for the BRI1-brassinolide complex. Reprints and permissions information is available at www.nature.com/reprints. The authors declare no competing financial interests.

Supplementary Material

Refer to Web version on PubMed Central for supplementary material.


Pierre-Alexandre Simon

Radu S Stoica

Frédéric Sur

An application of neural point processes to geophysical data

The huge amount of temporal data available nowadays in numerous scientific fields requires dedicated analysis and prediction methods. Stochastic temporal point processes are certainly one of the popular approaches available to model time series. While point processes have been successfully applied in many application domains, they need strong assumptions. For instance, the conditional intensity is often supposed to follow a particular parametric function, hence fixing a priori the structure of the events distribution: purely random or independent, clustered or regular. Recent papers investigate the use of models from machine learning dedicated to sequential events analysis, namely recurrent neural networks (RNN). These RNNs are expected to be versatile enough to automatically adapt to the data, without the need for a priori choosing the character of the events distribution. This paper presents a brief introduction to the so-called neural point processes and discusses numerical experiments. In particular, the presented real data application considers seismic data from the Guadeloupe region.

Introduction

Stochastic modelling is maybe one of the most used mathematical tools for geophysical data. Clearly, the main interest of using this modelling framework is to provide understanding and prediction of the phenomena underlined by the considered data. Let us consider in the following that our data is a collection of time occurrences of some geological events. This type of random sequence is called a temporal point process or simply a point process. A first question naturally arising is whether the events are independent, if they are clustered -that is one event may trigger another event, or if the events exhibit regularity or repulsion -an event does not favour that another event occurs immediately after. Once these hypotheses are tested, a particular model can be fitted to the data, and statistical inference can be performed, using the analytical properties and simulation algorithms of the considered process. The key tool of this approach is the conditional intensity. This quantity may be interpreted as the probability that a new event occurs knowing the observed process configuration. Important families of such stochastic processes are fully characterised by the conditional intensity. Among them we mention the Poisson processes, the Hawkes processes and the self-correcting processes. The Poisson processes allow to characterise independent events, the Hawkes process describe rather clustered events, while the self-correcting processes are related to repulsive processes. The simulation algorithms and the inference procedures of these processes is based on the computation of the conditional intensity. For a detailed mathematical presentation of point processes theory and of these particular processes, we recommend the monograph [START_REF] Daley | An introduction to the theory of point processes, volume 1 : Elementary theory and methods[END_REF].

The main drawback of using this approach is that in order to fit a model to the data, the type of the model should be chosen. Even if mixture modelling is chosen, fundamentally this question cannot be avoided.

Very recent works -Mei and Eisner (2017); Omi, Ueda, and Aihara (2019) -propose to use neural point processes to analyse time occurrences from seismic data, financial time series, or textual data. These authors used a Recurrent Neural Network (RNN) to model the conditional intensity of a point process. The great interest of adopting neural nets based inference if compared with classical statistical analysis is that no choice for the model family should be done.

The aim of this work is to apply and to extend the ideas presented in Omi, Ueda, and Aihara (2019) and to test them for predicting seismic data.

The structure of the paper is the following. Next section introduces fundamental facts regarding point processes and the conditional intensity. After this, the recurrent neural nets and neural point processes are described. The fourth section of the paper is dedicated to the obtained results on simulated and real data. The results on the simulated data show how the neural point process are able to characterise independent, clustered and repulsive point processes. The real data application considers seismic data from the Guadeloupe region. Finally, conclusions and perspective are depicted.

Temporal point processes

In the following, we consider temporal point process as an ordered sequence of time occurrences (i.e. random events) on the positive real axis. The realisation of the process is represented by {t 1 , t 2 , . . . , t n } with t i ∈ R + . Furthermore, the sequence is supposed to be locally finite, that is the number of events in a finite interval is always finite. We are interested in modelling the distribution of these events, after its observation in the interval [0, T ].

Modelling the conditional intensity

Three models of temporal point processes will be presented. Each of these models is able to describe a particular relation among the time events. Hence, the considered processes are the homogeneous Poisson processes (HPP), the self-exciting processes (SEP) also known as Hawkes processes, and the self-correcting processes (SCP), modelling independence, excitation, and inhibition between the occurrence of events, respectively. The conditional intensity function at time t is the rate of event occurrence as a function of the history H tn of the n event times preceding t. It is written as:

λ(t|H tn ) = lim ∆t→0 E(N ([t, t + ∆t))|H tn ) ∆t
where N ([t, t + ∆t)) is the number of event occurrence in the time interval [t, t + ∆t) and E(N ([t, t + ∆t))|H tn ) denotes the expected number of event occurrence in this interval, knowing the history of event times before time t. It is possible to describe the density function f (t|H tn ) (resp. the cumulative distribution function F (t|H tn )) of the next arrival event after the time t n as a function of the conditional intensity function λ(t|H tn ), by solving the following integral equation (Daley & Vere Jones, 2008, Sec. 14.3.8 p. 399):

λ(t|H tn ) = f (t|H tn ) 1 -F (t|H tn ) This gives f (t|H tn ) = λ(t|H tn )e -t tn λ(s|Ht n )ds (1)
and F (t|H tn ) = 1 -e -t tn λ(s|Ht n )ds .

(2)

For the HPP process, the conditional intensity is

λ(t|H tn ) = µ 0 (3)
with µ 0 ≥ 0 a constant. Note that the case µ 0 = µ 0 (t) corresponds to independence of events as well. In this paper, we have considered for simplicity only the case where µ 0 is a constant. This case guarantees the same frequency of events in the time interval [t, t + dt) independently of t.

An application of neural point processes to geophysical data Pierre-Alexandre Simon, Radu S. Stoica, Frédéric Sur In an SEP , the conditional intensity λ is often modelled as the sum of a constant background intensity and a function that combines exponential decays. This last component reflects the excitation involved by past events:

λ(t|H tn ) = µ 0 + α n i=1 e -β(t-t i ) (4)
with α, β > 0 and α < β. If β → ∞ a SEP tends to a HPP. Otherwise, the intensity encourages the occurrence of a new event t rather close to its ancestors. Hence, this type of process, depending on its parameters, tends to produce clustered configurations of events. Figure 1 illustrates the realisation of an HP, an SEP, and an SCP. Time events are marked as dots on the horizontal axis. The conditional intensity function λ(t|H t -) is represented in red. In green, the associated counting process is represented. The counting process N (t) is the family of random variables or the stochastic process given by the sum of events in the interval [0, t). Note that the conditional intensity of an SEP is a stochastic quantity.

For an SCP, the conditional intensity tends to inhibit the occurrence of close events. Here, for this purpose, an increasing exponential function was considered λ(t|H tn ) = e ρt-α n i=1 1{t i <t} .

(5)

An application of neural point processes to geophysical data Pierre-Alexandre Simon, Radu S. Stoica, Frédéric Sur with α, ρ > 0 and 1 the indicator function.

Parameter estimation

In this section, a brief and intuitive presentation for the derivation of the likelihood of the considered point processes is given. For a thorough and detailed presentation please refer to (Daley & Vere Jones, 2003, Sec. 7).

The joint probability density of a sequence of n consecutive events may be written as

f (t 1 , t 2 , ..., t n ) = n i=1 f (t i |t 1 , t 2 , ..., t i-1 ) = n i=1 f (t i |H t i-1 )
By plugging ( 1) and ( 2) into the preceding equation, the likelihood L of n events occurring in the [0, T ] time interval can be expressed as a function of the conditional intensity:

L = n i=1 f (t i |H t i-1 ) (1 -F (T |H tn )) = n i=1 λ(t i |H t i-1 )e - t i t i-1 λ(s|H s -)ds e -T tn λ(s|H s -)ds = n i=1 λ(t i |H t i-1 ) e -T 0 λ(s|H s -)ds since t 0 = 0.
The log-likelihood thus writes:

log(L) = n i=1 log(λ(t i |H t i-1 )) - T 0 λ(s|H s -)ds. ( 6
)
The above notation H s -is denoting the history of the process at time s but excluding s. The parameters of the considered point processes (SEP or SCP) given by ( 4) or ( 5) are estimated by maximising the corresponding log-likelihood function. General results and explanations regarding the convexity of the likelihood functions may be found in [START_REF] Daley | An introduction to the theory of point processes, volume 1 : Elementary theory and methods[END_REF].

Model validation

After the observation of a sequence t 1 , . . . , t n assumed to be the realisation of a temporal point process, under the hypothesis of a particular chosen model, parameter estimation may be achieved by maximising the corresponding likelihood. The model choice should still be validated. This can be obtained via the results presented in (Daley & Vere Jones, 2003, Thm. 7.4.I and Prop. 7.4.IV). These results guarantee that the transformed point process given by Λ(t 1 ), . . . , Λ(t n ) with

Λ(t) = t 0 λ(s|H s -)ds (7)
is a unit-rate Poisson process. Hence, the inter-arrival times of the transformed process

Λ(t 2 ) -Λ(t 1 ), Λ(t 3 ) -Λ(t 2 ), . . . , Λ(t n ) -Λ(t n-1 )
follow an unit-rate exponential distribution. Clearly, plugging into the previous sequence the conditional intensity with the estimated parameters leads to a statistical sample which distribution can be assessed via a Kolmogorov-Smirnov (KS) test. Besides, since Poisson processes give independent inter-arrival times, the significance of the correlation between the transformed inter-arrival times can also be tested with the Ljung-Box (LB) statistical test.

An application of neural point processes to geophysical data Pierre-Alexandre Simon, Radu S. Stoica, Frédéric Sur

Neural point processes

The classical statistical approach for modelling time events using point processes requires the choice of a model. This is not always very convenient, since it requires statistical tests to be done a priori to the parameter estimation. The advantage of using this modelling framework is that it inherits the whole mathematical toolkit in order to rigorously simulate, to predict and to perform a posteriori inference. At a first glance, the modelling of time events via recurrent neural networks seem to exhibit less mathematical control than the classical statistical approach. Nevertheless, it has the advantage of not depending on the choice of a model, once the structure of the considered neural network is fixed. This section first presents the recurrent neural networks, then their application to modelling of sequential time events based on a construction similar to the conditional intensity of temporal point processes. Finally, the training procedure of such networks is explained.

Modelling the intensity function with a Recurrent Neural Network

Contrary to the classical models described in Section 1, which are based on a parametric formulation of λ(t|H t -), neural point processes are modelled with a Recurrent Neural Network (RNN). The past of the process taken into account in the intensity function is captured by the so-called internal memory (hidden states h i ) of the RNN at each time-step.

Figure 2 illustrates the RNN architecture and the following notations. Inter-arrival times τ i = t i -t i-1 are used for computing the hidden state h i (which is a vector) at each time-step. The hidden state for the i th time-step is given by:

h i = σ 1 (W h h i-1 + v τ τ i + b h ) (8)
where σ 1 denotes an activation function (commonly the hyperbolic tangent function), W h , v τ , and b h denote respectively the weight matrix shared between the hidden states, the weight vector for interarrival times, and a bias term. The last hidden state is then used for the estimation of the intensity at time t i+1 . For instance, an exponential form may be assumed for the intensity, such as:

λ(t i+1 |H t i ) = exp(v y h i + v t τ i+1 + b y ) (9)
where v y , v tau , and b y are respectively a weight vector, a weight parameter, and a bias, h i a hidden state, and τ i+1 the i + 1-th inter-arrival time.

The output of the network at the i th time-step is eventually given by:

y i+1 = log(λ(t i+1 |H t i )) = v y h i + v t τ i+1 + b y (10)
An application of neural point processes to geophysical data Pierre-Alexandre Simon, Radu S. Stoica, Frédéric Sur

Modelling the integral of the intensity function

In RNN, the weights are set during the learning stage (see below) by maximising the log-likelihood function given by Equation ( 6). However, computing the log-likelihood requires to estimate the integral of the conditional intensity λ(t|H t -) with numerical methods as in [START_REF] Mei | The neural Hawkes process: A neurally self-modulating multivariate point process[END_REF]. Such an approach has a cost in terms of accuracy and computational time. A novel method developed in Omi, Ueda, and Aihara (2019) consists in estimating this integral by giving as an input of a feed-forward neural network the internal state of the RNN modeling the history H t , together with time t. The reason behind is that a well-suited feed-forward neural network should be able to learn the integral of λ(t|H t -) from samples of t and H t , since these networks satisfy the universal approximation theorem, see [START_REF] Cybenko | Approximation by superpositions of a sigmoidal function[END_REF]; [START_REF] Hornik | Approximation capabilities of multilayer feedforward networks[END_REF]. The intensity function is subsequently derived from the integral estimated by the feed-forward network by automatic differentiation. The corresponding neural architecture is described in Figure 3. At time-step i + 1, the output of the network combining the RNN and feed-forward network is the integral of the conditional intensity function between t i and t i+1 such as:

y i+1 = t i+1 t i λ(s|H s -)ds = Λ(t i+1 ) -Λ(t i )
In a similar way as for intensity function, the first part of the network (RNN) is providing a compressed representation of the history H t i via the last hidden state h i :

h i = σ 1 (W h h i-1 + v τ τ i + b h )
Then, the second part of the network (the feed-forward network) is modelling the integral of the intensity function taking as input the last hidden state of the RNN and the elapsed time since last event τ i+1 such as the output is:

y i+1 = v o σ l 2 (W l σ l 1 (W y h i + v t τ i+1 + b y ) + b l ) + b o
where W y and b y are respectively the weight matrix and the bias of the connection between the last hidden state h i and the first feed-forward network layer, and v t the weight vector of the connection between the i th inter-arrival and the first feed-forward network layer. W l and b l are respectively the weight matrix and the bias of the connection between the first and second feed-forward network layer and the vector v o between the second feed-forward network layer and the output y i+1 . σ l 1 and σ l 2 are respectively the activation functions used for the first and the second layer of the feed-forward network.

An application of neural point processes to geophysical data Pierre-Alexandre Simon, Radu S. Stoica, Frédéric Sur In all experiments discussed in Section 3, the feed-forward network architecture consists in two hidden layers including each 64 neurons. Its input layer consist in the last hidden state h i of the RNN and the time elapsed since last event occurrence t. The output layer consists in one neuron, giving a scalar value representing the integral of the intensity function between times t i and t i+1 .

Neural network training

As in any neural approach, the weights of the network are set according to a data set during the socalled training stage. The sequence of inter-arrival times is split into training and testing sets for the training and evaluation of the network, respectively. In machine learning application, it is important to evaluate models on data-sets which are not involved in training. During the training phase, the log-likelihood function ( 6) is maximised to derive weights and bias for the neurons of the network, similarly to the standard parametric model. With the notations of Figure 3, the log-likelihood writes:

log(L) = i log ∂ y i+1 ∂t -y i+1
The Back-Propagation Trough Time algorithm is used to obtain the gradient of the log-likelihood function. It consists in unfolding the RNN (the unfolded network is represented on Figure 2) and apply the back-propagation to this unfolded network. The neural network is fed consecutively with the past inter-arrival times considering a moving window of length M W . Considering a fixed-size moving window (hence a limited history) prevents the exploding gradient problem which makes learning to fail. For example, at time step i + 1, the RNN is fed with the sequence {τ i-M W , τ i-M W +1 , ..., τ i }. Then, the last hidden state vector h i provides a compressed representation of this history to the feedforward network modelling the integral of the intensity function at time t = τ i+1 . Finally, the network provides the output value y i+1 , which represents the integral of the intensity function between time t i and t i+1 .

Experiments

Simulated data

To experiment neural point processes, we generated 100,000 artificial events data from the following point processes, defined by their conditional intensity function λ * :

• Homogeneous Poisson process (µ 0 = 0.2)

• Self-exciting (Hawkes) process (µ 0 = 0.2, α = 0.8, β = 1.0)

• Self-correcting process (α = 1.0, ρ = 1.0) These processes were chosen as they are representative of typical practical situations.

For each process simulated, we computed as well values for the intensity function, its integral and the log-likelihood at each of the arrival times for comparative purposes.

Figure 4 shows the conditional intensity function λ derived by neural point process modeling, together with the ground-truth λ. For assessing the quality of fit, we computed different metrics such as the Mean Absolute Error (MAE), Bias, Mean Squared Error (MSE) and Root Mean Squared Error (RMSE) between the intensity evaluated at each new arrival and effective value. The Bias is indicating if we are over(resp. under)-estimating the effective intensity:

Bias = 1 n n i=1 λ(t i |H t i-1 ) -λ(t i |H t i-1 )
An application of neural point processes to geophysical data Pierre-Alexandre Simon, Radu S. Stoica, Frédéric Sur The RMSE penalises heavily significant differences on the intensity function evaluation (by squaring them), which is the main challenge for the Hawkes model when clusters of points form.

RMSE = 1 n n i=1 λ(t i |H t i-1 ) -λ(t i |H t i-1 ) 2
The RMSE between the estimation and the real intensity is lower than 10% of the mean intensity (equal to 5), showing that neural point process fits well the challenging Hawkes process.

We also computed the Kolmogorov-Smirnov p-value and Kullback-Leibler and Jensen-Shanon divergences for verifying the adequacy of the residual with the unit exponential distribution (as explained in Section 1.3), see Table 1. In the three typical cases, the p-value of Kolmogorov-Smirnov test are An application of neural point processes to geophysical data Pierre-Alexandre Simon, Radu S. Stoica, Frédéric Sur higher than the risk confidence level α = 0.05, which indicates that the null-hypothesis (adequacy of the empirical distribution with the unit-rate exponential distribution) cannot be rejected. The small values for KL and JS divergences are confirming the goodness-of-fit of the neural point process for each kind of simulated data. In a similar way, the p-value of Ljung-Box statistical test are higher than the risk confidence level for each case, indicating that the null-hypothesis (inter-arrival are not auto-correlated) cannot be rejected.

Metrics

Real data

We also proposed to experiment neural point processes on a real data-set gathering occurrence times for 2,264 earthquakes in Guadeloupe between 2004 and 2005 collected by the RING team from Geo-Ressources (see also [START_REF] Allal | Hawkes point processes based inference applied to seismic data analysis[END_REF]), illustrated in Figure 7. This data-set is divided into a training set (incl. 1,811 arrival times) and a validation set (incl. 453 arrival times). As Hawkes model is popular for modelling earthquake events, we proposed to compare the neural network model with a specific parametric Hawkes model whose parameters (cf Equation ( 4)) were estimated with the maximum likelihood method on the training set.

After training the network (with a moving window of 10 elements, as mentioned in Section 2.3), the intensity function (shown in Figure 8), its integral and the log-likelihood functions are computed at each occurrence time of the testing set for both the neural point process model (green) and the parametric Hawkes model (salmon). At a first glance, we observe in Figure 8 that the intensity is capturing well the excitation phenomenon both for parametric and neural point process models. We An application of neural point processes to geophysical data Pierre-Alexandre Simon, Radu S. Stoica, Frédéric Sur observe that the neural point process over-estimates the intensity peaks due to excitation effects. This is due to the few amount of data at disposal for training, when a larger amount would provide a smoother estimation as in the parametric model.

From the intensity, we are able to predict the occurrence of the next earthquake given the observation of the last 10 earthquakes using (1). We used the bisection method for neural point process as in Omi, Ueda, and Aihara (2019). It consists in using the property that {∀t > t i , Λ(t) -Λ(t i ))} follows a unit-rate exponential distribution over the interval [t i , t i+1 ]. The predicted next arrival time t i+1 is then defined as the median value and is obtained by solving Λ( t i+1 ) -Λ(t i ) = log(2). Practically, we computed the integral of the intensity function from the same inter-arrival input sequence, different value for t i+1 until the preceding condition is fulfilled. For parametric Hawkes An application of neural point processes to geophysical data Pierre-Alexandre Simon, Radu S. Stoica, Frédéric Sur processes, we fitted the model with training data and obtained the following parameters estimators ( µ 0 = 0.11, α = 0.09, β = 0.13). Then, we used the Ogata algorithm to simulate 100 inter-arrival times and take the median of the obtained distribution. We then compared these predicted arrival times with the real earthquake arrival times of the data-set. The different evaluation metric for both model are described in Table 2 We observe that the Mean Absolute Error rises up to approximately 4 hours with a positive bias indicating that both models are under-estimating the next earthquake arrival time of 4 hours in average. In other words, the next earthquake is occurring in average 4 hours after the time predicted, which is satisfying for safety purposes. Figure 9 shows a comparison between these results and the real events occurrence times. We can see that short inter-arrival times characterising the excitation phenomenon are well estimated by both models, which is the major challenge when predicting next event occurrence. It is interesting to note that the neural network performs similarly to the parametric Hawkes model on all metrics. However, while the parametric model requires the full occurrence time history, the neural network model has only been trained considering the last 10 earthquakes occurrence times (because of the moving window of length 10). Moreover, once the network has been trained, it only needed a few seconds to provide the next arrival prediction sequence where the standard parametric model required 15 minutes. These results are really encouraging since no specific intensity function form is required in the neural point process model. The modest computational resources are also worth mentioning.

An application of neural point processes to geophysical data Pierre-Alexandre Simon, Radu S. Stoica, Frédéric Sur

Conclusion

Within the neural point process model, we experimented the approach of Omi, Ueda, and Aihara (2019) for modelling the integral of the intensity function of temporal point process with compound neural network. We have shown through numerical experiments on synthetic and real data that neural point processes permit to model in a satisfactory manner three typical situations related to time events: independence, clustering and regular distributions. Compared with traditional point process approaches, no prior assumption on the intensity function is required, the weights of the neural networks automatically fitting the input data in the training stage. Nevertheless, the architecture of involved networks and their hyper parameters as well had to be tuned after several trials and errors.

  Figure 1: a: realisation of an homogeneous Poisson process (µ 0 = 1.0). b: realisation of a self-exciting (Hawkes) process (µ 0 = 1.0, α = 0.6, β = 0.8). c: realisation of a self-correcting process (ρ = 1.0, α = 1.0).

Figure 2 :

 2 Figure 2: A recurrent neural network.

Figure 3 :

 3 Figure 3: Architecture of the overall network.

Figure 4 :

 4 Figure 4: Neural network estimation of intensity function (green) against intensity function of Poisson process generating artificial data (black).

Figure 5 :

 5 Figure 5: Neural network estimation of intensity function (green) against intensity function of Hawkes process generating artificial data (black).

Figure 6 :

 6 Figure 6: Neural network estimation of intensity function (green) against intensity function of self-correcting process generating artificial data (black).

Figure 7 :

 7 Figure 7: Earthquake occurrence times and magnitudes.

Figure 8 :

 8 Figure 8: Derived intensity functions from testing set. The neural network estimation is in green and the standard parametric estimation is in salmon.

Figure 9 :

 9 Figure 9: Next arrival prediction on the testing set. The neural network estimation is in yellow and the standard parametric estimation is in salmon. Actual next arrival time is in black.

Table 1 :

 1 Evaluation metrics for each neural point process models

		Poisson H Self-correcting	Hawkes
	RMSE λ * (t)	0.01	0.08	0.47
	Bias λ * (t)	0.0	0.03	0.17
	KS p-value	0.48	0.12	0.38
	KL divergence	-0.001	0.018	-1.1.10 -3
	JS divergence -1.49.10 -5	-4.3.10 -4	-5.07.10 -6
	LB p-value	0.32	0.77	0.12

Table 2 :

 2 . Evaluation metrics for next arrival time.

	Metrics Hawkes Neural point process
	MAE	2.78	3.57
	Bias	1.28	2.22
	MSE	17.09	29.32
	RMSE	4.13	5.42

Acknowledgments

The authors would like to thank the RING team (GeoRessources lab) for kindly providing the seismic data.


Bernhard Hohmann

Mérouane Debbah
email: merouane.debbah@supelec.fr

Andreas Kröpfl
email: a.kroepfl@zydacron.com

Reliable Multimedia Transmission in Wireless Ad-Hoc Networks for Telehealth Systems

This paper proposes a time slotted distributed protocol to enhance transmissions in ad-hoc networks. This is achieved by employing the well known cooperative virtual multiple input multiple output (MIMO) technique. A general protocol for N cooperating nodes is presented and evaluated considering bit error rate (BER), signal to noise ratio (SNR) and delay as target metrics. We show in particular that there is a trade off between performance, complexity and delay. Interestingly, reliable multimedia transmission can be achieved with reasonable delay constraints, as required in many telehealth applications. 1

I. INTRODUCTION

Ad-hoc networks are gaining more and more importance, due to their flexibility and ease of deployment. However, the problem of fading is crucial in wireless communications and one needs to improve the reliability of these networks. One way of doing that is to apply diversity techniques such as multiple input multiple output (MIMO) [START_REF] Telatar | Capacity of multi-antenna gaussian channels[END_REF]. These systems require, however, multiple antennas at the sender and/or at the receiver. If the nodes do not have multiple antennas due to cost or power constraints, there is the possibility to employ virtual MIMO or cooperative diversity schemes as shown for example in [START_REF] Laneman | Cooperative diversity in wireless networks: Efficient protocols and outage behavior[END_REF]. In this paper, we will focus our attention on two specific techniques of cooperation, namely amplify and forward (AF) and decode and forward (DF) [START_REF] Laneman | Cooperative diversity in wireless networks: Efficient protocols and outage behavior[END_REF]. In order to improve the performance of the DF scheme, we derive an extension of the Alamouti scheme that was introduced in [START_REF] Alamouti | A simple transmit diversity technique for wireless communications[END_REF]. This scheme is a simple diversity technique for two transmitting and one or two receiving antennas which achieves full diversity. This paper, however, extends the approach to a virtual Alamouti scheme with N cooperating nodes. One key feature of this scheme is that only linear processing at the receiver is involved. This makes this scheme very attractive for receivers with power constraints. The primary application of the developed protocol is for telehealth applications which are becoming more and more important (see for example [START_REF] Keshavarz | Distributed vision-based reasoning for smart home care[END_REF], [START_REF] Williams | Distributed smart cameras for aging in place[END_REF]). One particular example of interest for the authors are homecare scenarios with distributed interactive terminals 2 and monitoring cameras, where it is interesting to use adhoc networking. In such scenarios, multimedia data has to be transmitted reliably and the proposed scheme provides new 1 This work is supported in part by the COAST [START_REF] Coast | www.coast.at[END_REF] project group. 2 See for example the Zydacron Carestation, www.zydacron.com possibilities in the field. The paper is organized as follows.

Section II introduces the system model and in Section III the protocol is described in detail. Section IV discusses simulation results and finally, Section V summarizes the paper with some concluding remarks.

II. SYSTEM MODEL

In the proposed system model, N nodes intend to cooperate in order to transmit data reliably. In the simulations, no coding delays are considered and the nodes are assumed to be synchronized. Furthermore, we consider a flat fading (which can be achieved by using an OFDM type technology) additive white gaussian noise (AWGN) channel which stays constant over at least one joint transmission of the participating nodes. The normalized transmit energy is fixed to 1. The general channel model for a two way communication follows below, where y is the received symbol, x is the transmitted symbol, h is the channel and n is a zero mean gaussian noise with unit variance.

y = hx + n (1)
A graph illustrating the cooperation between the nodes is given in Fig. 1. In the graph, S, D and R i represent the sending node, the destination node and the relays, respectively. The naming of the corresponding channels is as well included in the graph, e.g. the channel from the second relay R 2 to the destination D is called h 2D . The total received signal at the destination D for N cooperating nodes (i.e. N -1 relays) for a DF scenario and without any time slotted protocol is given by

y = h SD x + N -1 i=1 h iD x + n (2)
(where the signal is decoded first at each relay and then forwarded). In this case, every relay first perfectly decodes x and then forwards it toward the destination. In the AF case the received signal at the relay is simply forwarded. Thus, it forwards h Si x + n Si , which is the received signal at the relay R i . Hence, also the noise n Si gets amplified and the total received signal at the destination D is given by

y = h SD x + N -1 i=1 h iD h Si x + h iD n Si + n (3) R 1 R 2 R 3 S D h 1D h S1 h S2 h S3 h 2D h 3D h SD Fig. 1.
System model, where S, D and R i represent the sending node, the destination node and the relays, respectively. The channels are labeled according to the nodes they are connecting, thus, the channel from the second relay R 2 to the destination D is called h 2D .

TABLE I TIME SLOTTED STRUCTURE FOR AMPLIFY AND FORWARD

t 1 t 2 t 3 t 4 • • • t N S x 1 R 1 x 1 R 2 x 1 R 3 x 1 . . . . . . R N -1 x 1 D y 1 y 2 y 3 y 4 • • • y N
Unfortunately, as one can see, if all the relays send simultaneously, inter-symbol interference appears and the symbols can only be decoded by a computationally expensive detector. In order to avoid this, we apply a time slotted protocol.

III. PROTOCOL

The protocol is based on a time slotted structure where the sender picks a predefined number of N -1 participating relays out of the nodes within reach. As two different strategies are implemented, the protocol will be described in the following two sections, describing first the AF strategy in Section III-A and then DF in Section III-B.

A. Amplify and Forward

The AF strategy [START_REF] Laneman | Cooperative diversity in wireless networks: Efficient protocols and outage behavior[END_REF] is simple. In the first time slot the sender broadcasts a symbol to all the relays and in the following time slots all the relays transmit it one after the other. The symbols can then be decoded by a maximum likelihood (ML) decision, without inter-symbol interference for which the complexity is only related to the size of the constellation (and not the number of relays as previously). Thus, full diversity is achieved at the expense of a non-negligible delay. The time slotted structure is illustrated in Table I.

The quality of the received signal strongly depends on the energy of the channels to and from a relay, i.e., |h Si | 2 and |h iD | 2 , as they are amplifying or attenuating the signal. Hence, the decision on which N -1 nodes are chosen as relays is based on these energies. In case of one relay, the optimal node is the one which maximizes the equation below, as it expresses the energy E, received at the destination.

E = |h Si | 2 |h iD | 2 σ 2 (1 + |h iD | 2 ) ( 4
)
where σ is the variance of the noise. The optimal solution for choosing N -1 relays is straightforward and given by the following metric

E = |h Si | 2 |h iD | 2 + |h S(i+1) | 2 |h (i+1)D | 2 + • • • σ 2 ((N -1) + |h iD | 2 + |h (i+1)D | 2 + • • • • • • + |h S(N -1) | 2 |h (N -1)D | 2 • • • + |h (N -1)D | 2 ) (5)
In this formula, all possible combinations of the available set of nodes have to be considered. This would be computationally very expensive. Thus, all available nodes are one after another set in Equation 4and ranked by E. After that, the best N -1 nodes are chosen in the order of maximizing the equation, which still gives good results.

B. Decode and Forward

In case of DF, the sender S broadcasts two symbols x 1 and x 2 to an odd number (N -1) of relays in the first two time slots t 1 and t 2 . Then the relaying nodes first have to decode the received signal and then reencode it. Let us first consider a simple example of one relay R 1 helping the source S to transmit these symbols. Applying the Alamouti scheme [START_REF] Alamouti | A simple transmit diversity technique for wireless communications[END_REF] here, the symbols have to be encoded according to Table II, in time slots t 3 and t 4 . The destination node now has just to perform a linear combination of the two received symbols y 1 and y 2 and the channels h 1D and h SD , which is computationally inexpensive and, thus, energy efficient. The linear combinations follow below

ŷ1 = h * SD y 1 + h 1D y * 2 = x 1 (|h 1D | 2 + |h SD | 2 ) + n (6) ŷ2 = h * 1D y 1 -h SD y * 2 = x 2 (|h 1D | 2 + |h SD | 2 ) + n (7)
The linear combinations ŷ1 and ŷ2 give an amplified version of the original symbols x 1 and x 2 respectively. By a ML decoding the estimates of x 1 and x 2 can be retrieved. It has to be noted that perfect channel knowledge is assumed. In case of more than one relay, the time slotted structure which is shown in

• • • t N +1 t N +2 S x 1 x 2 x 1 -x * 2 R 1 x 2 x * 1 R 2 x 1 -x * 2 R 3 x 2 x * 1 . . . . . . R N -2 x 1 -x * 2 R N -1 x 2 x * 1 D y 1 y 2 y 3 y 4 y 5 y 6 • • • y N +1 y N +2
different routes, a diversity gain to overcome deep fades is achieved.

The strategy of choosing the relay nodes is slightly simpler here. In this scheme we assume that it is sufficient to consider the energy of the channel between sender and relay, i.e., |h Si | 2 . This channel is of course more important than the channel between relay and destination, as the relay has to reliably decode the symbol. Otherwise the relay could forward a wrong symbol and, thus, degrade the joint transmission. Therefore, nodes are simply chosen in order of their channel energies |h Si | 2 .

IV. ANALYSIS

Simulations have been performed for BPSK and QPSK. The performance of AF and DF in terms of BER vs. SNR is analyzed. Furthermore, the delay and the node selection strategy are evaluated.

A. BER vs. SNR

The results for AF and DF using BPSK encoding are presented in Fig. 2 and Fig. 3. It can be observed, that in the two node scenario, i.e., one relay, AF is performing better than DF. This is due to the optimum relay selection technique of AF in case of one relay. However, employing more relays, DF outperforms AF significantly. For example, DF with 6 nodes gives better results than AF with 8 nodes over the whole simulated SNR range. Furthermore, in low SNR regions DF performs better than AF at any number of nodes.

In case of QPSK, shown in Fig. 4 and Fig. 5, the advantage of DF over AF increases. Here, the supremacy of AF in case of one relay becomes vanishingly small. Furthermore, DF with 4 nodes outperforms the 8 node AF simulation. However, much more nodes have to cooperate, to achieve low BERs at a given SNR, when employing QPSK constellations.

B. Delay

By increasing the number of cooperating nodes the performance increases likewise. Theoretically, it would be possible to reach almost any BER at a given SNR. This, however, is in most cases very unrealistic, because the delay grows as well significantly. Due to space constraints, but without loss of concision, only BPSK results are presented for the delay and node selection performance analysis. In Fig. 6, the delay for BPSK in time slots vs. BER at SNR = 8 is presented. In DF, in one cooperative transmission always two symbols are transmitted jointly due to the coding scheme. In contrast to AF where always just one symbol is treated. In order to be able to compare AF and DF, the delay in time slots is drawn for the transmission of one symbol for both AF and DF. It can be observed that DF performs better in means of delay. DF is faster and reaches much better BER, except for the two node scenario, where AF and DF have the same delay of two time slots. The superiority of DF over AF becomes again much more obvious with QPSK. Here the BER curve flattens out very fast for AF, whereas for DF it still steadily decreases at 10 nodes. However, coding delays which have higher impact on DF than AF are not considered. It is interesting to note that the DF curve with BPSK constellations seems to start flattening out at 10 nodes as it is already in a low BER region of 10 -5 , whereas the DF curve in QPSK is still quite steep and only in the 10 -3 BER region. This indicates that there might be a potential if one wants to use a great number of relays with higher order M-PSK constellations.

C. Node Selection

In order to prove the efficiency of the employed node selection strategy by simulation, a comparison with random node selection has been performed. In Fig. 6 the dash dotted graphs show the BER curves for random node selection. It can be observed that the node selection strategies for AF and DF result in a significant gain. Furthermore, the result of the optimal node selection strategy for AF in case of one relay can be observed when using BPSK. When employing QPSK constellations, the optimal node selection strategy for one relay equals out with the generally better performance of DF. Thus, the curves start at the same point.

V. CONCLUSION

In this paper we presented a new scheme for cooperative transmission in ad-hoc networks which achieves full diversity. It is shown that by employing the well known Alamouti scheme in DF manner or even only an AF strategy, with just one relay the performance can be improved significantly. In this case AF might be preferable due to its simple implementation structure. However, the good performance in this case results mainly from the complex node selection strategy where all channels have to be known. The node selection strategy for DF on the other hand relies just on the knowledge of the channels from the sender to the relays, so it might be more realistic to implement. When employing more than just one relay, DF gives better BER performance at lower delays. Still, the delay increases linearly with the number of relays and, thus, high numbers of relays would usually not be preferable. Hence, in case of power constraints and high reliability demands, the proposed virtual MIMO protocol is shown to be the right approach. Future work will be on extending the scheme by employing other space-time block codes [START_REF] Tarokh | Space-time block codes from orthogonal designs[END_REF].

Fig. 2 .

 2 Fig. 2. The BER vs. SNR for AF (BPSK).

Fig. 3 .

 3 Fig. 3. The BER vs. SNR for DF (BPSK).

Fig. 4 .

 4 Fig. 4. The BER vs. SNR for AF (QPSK).

Fig. 6 .

 6 Fig.6. The delay in time slots vs. BER for AF and DF using BPSK encoding at SNR = 8. The dash dotted lines show the curves obtained with the same input parameters, but without any node selection strategy.

Fig. 5 .

 5 Fig. 5. The BER vs. SNR for DF (QPSK).