Denise Martinez-Zapien

Francesc Xavier Ruiz

Juline Poirson

André Mitschler

Juan Ramirez-Ramos

Anne Forster

Alexandra Cousido-Siah

Murielle Masson

Scott Vande Pol

Alberto Podjarny

Gilles Travé
email: gilles.trave@unistra.fr

Katia Zanier
email: zanier@unistra.fr

Structure of the E6/E6AP/p53 complex required for HPVmediated degradation of p53

The p53 pro-apoptotic tumor suppressor is mutated or functionally altered in most cancers. In epithelial tumors induced by "high-risk" mucosal Human Papillomaviruses (hrm-HPVs), including human cervical carcinoma and a growing number of head-and-neck cancers 1 , p53 is degraded by the viral oncoprotein E6 2 . In this process, E6 binds to a short LxxLL consensus sequence within the cellular ubiquitin ligase E6AP 3 . Subsequently, the E6/E6AP heterodimer recruits and degrades p53 4 . Neither E6 nor E6AP are separately able to recruit p53 3 , 5 , and the precise mode of assembly of E6, E6AP and p53 is unknown. Here, we solved the crystal structure of a ternary complex comprising full-length HPV16 E6, the LxxLL motif of E6AP and the core domain of p53. The LxxLL motif of E6AP renders the conformation of E6 competent for interaction with p53 by structuring a p53-binding cleft on E6. Mutagenesis of critical positions at the E6-p53 interface disrupts p53 degradation. The E6-binding site of p53 is distal from previously described DNA-and protein-binding surfaces of the core domain. This suggests that, in principle, E6 may avoid competition with cellular factors by targeting both free and bound p53 molecules. The E6/ E6AP/p53 complex represents a prototype of viral hijacking of both the ubiquitin-mediated protein Users may view, print,

degradation pathway and the p53 tumor suppressor pathway. The present structure provides a framework for the design of inhibitory therapeutic strategies against HPV-mediated oncogenesis.

Papillomaviruses are small DNA viruses, which infect the mucosal and cutaneous epithelia of most vertebrate species. HPV16 is the most prevalent and best studied hrm-HPV, responsible for 50% of cervical carcinomas and for most HPV-positive head-and-neck cancers [START_REF] Bosch | Comprehensive control of human papillomavirus infections and related diseases[END_REF] . The HPV oncoproteins E6 and E7 recognize numerous host proteins, in large part by hijacking cellular domain-motif interaction networks [START_REF] Davey | How viruses hijack cell regulation[END_REF] . In particular, most mucosal and cutaneous E6 proteins recognize cellular acidic leucine(L)-rich LxxLL motifs (reviewed in [START_REF] Pol | Papillomavirus E6 oncoproteins[END_REF] ). In a recent structural study [START_REF] Zanier | Structural basis for hijacking of cellular LxxLL motifs by papillomavirus E6 oncoproteins[END_REF] , we have shown that LxxLL motifs bind to a conserved pocket of E6, which is contributed by the protein's N-and C-terminal zinc-binding domains (E6N and E6C) and helix linker.

In E6-mediated degradation of p53, hrm-HPV E6 proteins interact with the LxxLL motif of E6AP, leading to recruitment and polyubiquitination of p53. The isolated LxxLL peptide of E6AP (named e6ap from here on) is sufficient to render E6 liable to interact with p53 [START_REF] Ansari | Peptide interactions stabilize and restructure human papillomavirus type 16 E6 to interact with p53[END_REF] .

Furthermore, several studies indicate that the "core" (DNA binding) domain of p53 is required for the interaction with E6/E6AP 9 -11 . We thus proceeded to reconstitute a minimal E6/E6AP/p53 ternary complex in vitro (Extended Data Fig. 1). The solubility enhanced HPV16 E6 4C/4S mutant (named E6 from here on), which degrades p53 with wild-type efficiency [START_REF] Zanier | Solution structure analysis of the HPV16 E6 oncoprotein reveals a self-association mechanism required for E6-mediated degradation of p53[END_REF] , was assembled with e6ap (sequence E 1 L 2 T 3 L 4 Q 5 E 6 L 7 L 8 G 9 E 10 E 11 R [START_REF] Zanier | Solution structure analysis of the HPV16 E6 oncoprotein reveals a self-association mechanism required for E6-mediated degradation of p53[END_REF] ) fused to a crystallization-prone mutant of the maltose binding protein (MBP) [START_REF] Zanier | Structural basis for hijacking of cellular LxxLL motifs by papillomavirus E6 oncoproteins[END_REF] (Extended Data Fig. 2). The resulting E6/MBP-e6ap heterodimer (named E6/e6ap from here on) was found to interact with the isolated p53 core domain (residues 94-292, named p53core from here on) by gel filtration chromatography and isothermal titration calorimetry (K D = 22 μM) (Extended Data Fig. 3 and Extended Data Table 1). In vivo, this affinity of p53 for E6/E6AP is likely to be enhanced by avidity effects, since p53 is tetrameric and E6AP can form trimers [START_REF] Ronchi | The active form of E6-associated protein (E6AP)/ UBE3A ubiquitin ligase is an oligomer[END_REF] .

The E6/e6ap/p53core ternary complex raised several crystals diffracting up to 2.25 Å resolution using synchrotron radiation. This allowed structure determination by molecular replacement (Fig. 1a and Extended Data Table 2). The asymmetric unit of the crystal comprises two E6/e6ap/p53core heterotrimers, which contact each other mostly via MBP and display nearly identical structures except for the relative orientation of the MBP moieties (Extended Data Fig. 4). The structures of p53core and E6/e6ap observed in the heterotrimers are superimposable with previous structures of p53core and of E6/e6ap heterodimer, except for residues 1-8 of E6 and 10-12 of e6ap, which change conformation upon p53 binding (Extended Data Fig. 5). The similarities between the structures of the two heterotrimers in the crystal and previously solved structures of separate elements suggest that MBP does not significantly alter the overall conformation of the E6/e6ap/p53core complex.

In each heterotrimer p53core binds to a cleft, which is formed by the E6N and E6C domains and held in place by contacts tethering the domains to the e6ap peptide (Fig. 1b and Fig 2a). The E6-p53 interface covers approximately 1200 Å 2 . The C-terminus of the e6ap peptide (residues 10-12, Extended Data Fig. 6a) also lies proximal to p53core (Extended Data Fig. 6b), but its structure is poorly defined, possibly due to an influence of the adjacent MBP tag. Nevertheless neither point mutations at residues 10-12 of e6ap nor extension of the e6ap peptide's C-terminal boundary altered p53 binding affinity (Extended Data Fig. 6c and Extended Data Table 1), suggesting that the e6ap C-terminus does not contribute significant intermolecular contacts to p53.

The E6-p53 interface can be divided into three sub-interfaces (Fig. 1a and Fig. 2b). Subinterface I is dominated by polar interactions and brings together residues in the N-terminal arm and α1 helix of E6 and residues of the N-terminal arm, β1 and β10 strands of p53core (Fig. 2b-c). E6 residues Glu7 and Glu18 establish a bi-dentate salt bridge with Lys101 of p53 as well as water-mediated interactions with other p53 residues. In particular, Glu18 contributes to a network involving side-chains and backbone groups of Gln14, Arg10 and Lys11 of E6, and of Thr102 and Asn268 of p53. Consistently, E6 mutations E18A and E18R impair ternary complex assembly (Fig. 3a) and p53 degradation (Fig. 3b). Furthermore, Gln104 and Gly105 of p53 hydrogen bond to the backbone of E6 residues Arg8 and Gln6, respectively, thus altering the conformation of the N-terminal arm of E6 (residues 1-8) (Extended Data Fig. 5).

Sub-interface II mainly involves the α2 helix of the E6N domain and the α2 helix of p53core, interacting through hydrophobic, charged and polar contacts (Fig. 2b-c). E6 residue Phe47 intercalates between p53 residues Ala129 and Arg290. In turn, Arg290 establishes a salt bridge with Asp44 of E6. These key interactions explain the reported dominant-negative phenotype of the E6 F47R mutant, which is defective for p53 degradation, restores high p53 levels and drives senescence [START_REF] Ristriani | A single-codon mutation converts HPV16 E6 oncoprotein into a potential tumor suppressor, which induces p53-dependent senescence of HPV-positive HeLa cervical cancer cells[END_REF] . Consistently, here we find that E6 F47R is defective for interaction with both p53core (Extended Data Fig. 3a) and fulllength p53 (Fig. 3a). Besides Phe47, other E6 residues in sub-interface II (Ile23, His24 and Tyr43) provide hydrophobic contacts, whereas E6 residue Asp49 establishes polar interactions with His115 in the L1 loop of p53. Mutations disrupting the Asp44-Arg290 and Asp49-His115 interactions (D44R and D49A in E6 and the R290E in p53) impair both p53 ternary complex assembly (Fig. 3a) and p53 degradation (Fig. 3b). By contrast, creation of a swapped Glu290-Arg44 salt bridge, by combining D44R E6 and R290E p53 mutants, partially restores p53 degradation (Fig. 3b).

Finally, sub-interface III encompasses hydrophobic interactions between E6C residues Leu100 and Pro112 and p53core surface residues Leu114 and Trp146, as well as a number of water-mediated backbone-to-backbone contacts (Fig. 2b-c). Notably, Leu100 and Pro112 of E6C are proximal to conserved Arg102, which shapes the p53 binding cleft by mediating crucial interactions with the E6N domain and the e6ap peptide [START_REF] Zanier | Structural basis for hijacking of cellular LxxLL motifs by papillomavirus E6 oncoproteins[END_REF] .

p53 degradation activity is a hallmark of hrm-HPV E6 proteins 15 -17 . The structure explains the critical contributions towards p53 degradation of particular residues, which are differentially conserved in high-versus low-risk mucosal (lrm) HPV E6. In sub-interface I, most conserved positions in hrm-HPV E6 proteins correspond to Phe2, Pro5, Glu7, Arg8 and Pro9 (Fig. 2a). While Glu7 mediates direct contacts to p53, Phe2, Pro5, Arg8 and Pro9 play an indirect role at the interface by shaping the conformation of the N-terminal region (Extended Data Fig. 7). Indeed, site-directed mutagenesis of Phe2, Pro5, Arg8 or Pro9 18 -20 impairs ternary complex assembly and p53 degradation activities. In sub-interface II, most conserved positions in hrm-HPV correspond to Phe47, Asp44 and Asp49 (Fig. 2a), which are here found to mediate crucial contacts to p53. These residues were formerly found to be important for p53 degradation 14 , 18 , 21 . Notably, Phe47 and Asp44 also participate in the dimerization of the E6N domain and subsequent in vitro self-association of the entire E6 protein [START_REF] Zanier | Solution structure analysis of the HPV16 E6 oncoprotein reveals a self-association mechanism required for E6-mediated degradation of p53[END_REF] . Therefore, E6 self-association and E6 binding to p53 are two distinct and competing processes mediated by partially overlapping interaction surfaces.

The LxxLL peptide of E6AP is absolutely required for E6 binding to p53, yet it does not contribute significant contacts to p53. Furthermore, in the absence of the LxxLL motif of E6AP, the few interactions connecting E6N, linker helix and E6C should be insufficient to maintain the overall E6 architecture observed in the E6/e6ap heterodimer structure (see [START_REF] Zanier | Structural basis for hijacking of cellular LxxLL motifs by papillomavirus E6 oncoproteins[END_REF] for further discussion). These observations indicate that the LxxLL motif structures the p53 binding cleft on E6, thereby rendering E6 competent for interaction with p53. Interestingly, we have shown that an in vitro selected peptide, targeting the LxxLL pocket of HPV16 E6, induces recruitment of p53 to E6 22 , 23 . Consequently, cellular proteins other than E6AP, which bind to the LxxLL pocket, might also promote the E6-p53 interaction.

p53core is both a DNA binding domain and a protein-protein interaction hub. E6 interacts with the N-terminal arm of p53core and one of the edges of the β-sandwich. Previous mutagenesis studies had already suggested a role for these regions in E6 binding 9 , 11 , 24 . The E6-binding interface on p53core is distal from both the DNA-binding region and proteinbinding surfaces observed in all other solved complexes of p53core (Fig. 4a). Indeed, previously described protein-binding interfaces of p53core all overlap, albeit at different extents, with the DNA-binding region. In particular, the Large T antigen (LTag) of oncovirus SV40, which does not degrade p53, buries the entire DNA binding interface of p53core, thereby inhibiting p53 trans-activation activity [START_REF] Lilyestrom | Crystal structure of SV40 large Tantigen bound to p53: interplay between a viral oncoprotein and a cellular tumor suppressor[END_REF] . Consistent with these observations, two frequent cancer-associated mutations (R273C and R273L), affecting a prominent DNAbinding arginine residue of p53core, abolish binding to SV40 LTag, but do not alter the interaction with E6/E6AP [START_REF] Scheffner | Interaction of the human papillomavirus type 16 E6 oncoprotein with wild-type and mutant human p53 proteins[END_REF] . Therefore, E6 might avoid competition with cellular factors by targeting both 'free' p53 as well as p53 bound to DNA or other proteins (Fig. 4b). This, along with the irreversible character of the degradation process, renders E6 a potent inactivator of p53 functions.

Recent studies employing pro-apoptotic peptides [START_REF] Zanier | The E6AP binding pocket of the HPV16 E6 oncoprotein provides a docking site for a small inhibitory peptide unrelated to E6AP, indicating druggability of E6[END_REF] as well as small molecules 27 , 28 directed against the LxxLL pocket have provided experimental evidence that this pocket is druggable. The p53-binding cleft observed in the present structure may represent a second potential binding site for drugs. Combinatorial strategies targeting both the LxxLL pocket and the p53-binding cleft could result in efficient disruption of the E6/E6AP/p53 complex.

Methods

Preparation of protein samples

DNA constructs-Residues 403 to 414 of human E6AP (peptide sequence

E 1 L 2 T 3 L 4 Q 5 E 6 L 7 L 8 G 9 E 10 E 11 R 12 )
were cloned via a three-alanine linker at the C-terminus of a mutant MBP used to promote crystallization [START_REF] Zanier | Structural basis for hijacking of cellular LxxLL motifs by papillomavirus E6 oncoproteins[END_REF] . The point mutations introduced in MBP (D83A, K84A, K240A, E360A, K363A and D364A) have been previously described to increase the propensity of MBP to crystallize [START_REF] Moon | A synergistic approach to protein crystallization: combination of a fixed-arm carrier with surface entropy reduction[END_REF] . The E6 4C/4S mutant of HPV16 [START_REF] Zanier | Solution structure analysis of the HPV16 E6 oncoprotein reveals a self-association mechanism required for E6-mediated degradation of p53[END_REF] (named E6 from here on) and the core (or DNA binding) domain of human p53 (residues 94 to 311) were cloned into the pETM-41 vector containing an N-terminal His 6 -MBP tag followed by a TEV cleavage site.

Protein expression and purification-MBP-e6ap, His 6 -MBP-tev-E6 and His 6 -MBPtev-p53core constructs were over-expressed separately in E. coli BL21(DE3) cells at 15 °C for 18 hours. All constructs were purified separately by amylose affinity chromatography in buffer A (50 mM Tris pH 6.8, 400 mM NaCl, and 2 mM DTT). To remove soluble aggregates, all affinity purified samples were ultracentrifuged at 110,000 g in a swing SW41 rotor (Beckman) for 16 hours at 4 °C. Hence p53core and E6 samples were digested by TEV protease. In the case of p53core, after TEV digestion samples were additionally purified on an heparin column. The resulting MBP-e6ap, E6 and p53 samples were thus concentrated and loaded separately onto a Superdex 75 HiLoad 16/60 gel filtration column (GE Healthcare) equilibrated in buffer A. All purification buffers were filtered, degassed and saturated with argon.

Crystallization

The E6/e6ap/p53core complex was reconstituted by mixing MBP-e6ap and E6 and p53core samples in a 1:1:1 stoichiometric ratio in buffer B (50 mM Tris pH 6.8, 200 mM NaCl, 2 mM DTT, 5 mM maltose) and concentrated to 18 mg/ml prior to crystallization. Crystallization conditions were screened using commercially available kits (Qiagen, Hampton Research, Emerald Biosystems) by the sitting-drop vapor-diffusion method in 96well MRC 2-drop plates (SWISSCI), employing a Mosquito robot (TTP Labtech). Initial crystals were obtained and used as seeds during further optimization steps.

After optimization, crystals (125 × 80 × 80 μm) grew in sitting drops made from 400 nl of protein solution at 18 mg/ml, 370 nl of reservoir solution containing 7.5 % polyethylene glycol (PEG) 20K, 50 mM MES (pH 6.5), and 30 nl of seeds. Drops were equilibrated against 50 μl of reservoir solution at 290 K. Crystals were sequentially transferred through two cryo solutions of reservoir solution supplemented with 15 % (v/v) and 25 % (v/v) of PEG 200, respectively. The crystals were flash-cooled and stored in liquid nitrogen.

Data collection and processing, and structure determination

X-ray diffraction data were collected on the X06DA beamline at the Synchrotron Swiss Light Source (SLS, Villigen, Switzerland). Data were acquired from a single cryo-cooled crystal (100 K) on a Pilatus-2M detector. 200° data were collected up to a resolution of 2.25 Å using 0.5° rotation and 0.8 sec exposure time with 20 % beam attenuation for each image. The data were indexed, processed and scaled using HKL2000 [START_REF] Otwinowski | Processing of X-ray diffraction data collected in oscillation mode[END_REF] . The crystals belong to the monoclinic space group P2 1 with unit cell parameters a=78.17Å, b=129.37Å, c=82.17Å and β=92.4°, with a refined crystal mosaicity of 0.28 -0.34°. The asymmetric unit contains two copies of the E6/MBP-e6ap/p53core heterotrimer, with a corresponding Matthew's coefficient of 2.48 Å 3 /Da and a solvent content of 50.34 %. The structure was solved by two sequential molecular replacements using Phaser [START_REF] Mccoy | Phaser crystallographic software[END_REF] and the structures of MBP, E6/e6ap complex and p53core as templates (PDB IDs 4GIZ and 1TUP, respectively). Crystallographic refinement involved repeated cycles of conjugate-gradient energy minimization and temperature-factor refinement, and was performed using PHENIX 32 followed by iterative model building in Coot [START_REF] Emsley | Coot: model-building tools for molecular graphics[END_REF] . All the disordered residues were fitted to the built electron density map, except from e6ap residue Arg12 in one of the two heterotrimers (corresponding to chain B residue Arg383 in the pdb file).

Data collection and refinement statistics are summarized in Extended Data Table 2. The quality of the refined models was assessed using MOLPROBITY [START_REF] Chen | MolProbity: all-atom structure validation for macromolecular crystallography[END_REF] . All molecular graphics figures were made using PyMOL [START_REF] Delano | Benchmarking a luciferase complementation assay for detecting protein complexes[END_REF] . E6 residues at interface regions (maintext Fig. 1b and Fig 2a ) were identified by the observation of an increase in solvent accessibility obtained upon removal of e6ap or p53core from the pdb file of the ternary complex. Positive "Δ exposure" values (maintext Fig. 2a) indicate E6 residues in atomic contact with or p53.

The refined model and the structure factor amplitudes have been deposited in the Protein Data Bank under the PDB code 4XR8.

DNA constructs for triple pull-down and p53 degradation assays

HPV16 E6 mutations were inserted in the background of the E6 4C/4S construct to prevent cross-linking of E6 molecules through intermolecular cysteine bridges. We have previously shown that E6 4C/4S and wt HPV16 E6 have indistinguishable p53 degradation activities in vitro [START_REF] Zanier | Solution structure analysis of the HPV16 E6 oncoprotein reveals a self-association mechanism required for E6-mediated degradation of p53[END_REF] . p53 mutations were inserted in the background of wild-type full-length human p53.

All DNA constructs were verified by sequencing.

Triple pull-downs-DNA constructs encoding for full-length HPV16 E6 and p53, and a large fragment of E6AP (residues 291-875 of isoform II) were inserted into the final destination vectors by Gateway cloning (Invitrogen). HPV16 E6 and E6AP were inserted into the GPCA pSPICA-N2 and pSPICA-N1 vectors respectively. GPCA vectors pSPICA-N1 and pSPICA-N2 (both derived from the pCiNeo mammalian expression vector) allow expression of test proteins as fusions to the C-terminus of Gluc1 and Gluc2 complementary fragments of the Gaussia princeps luciferase, respectively 36 . By contrast p53 constructs were inserted in the BioEase-DEST vector 36 , which incorporates a 72-amino acid sequence from Klebsilla pneumonia that directs in vivo biotinylation.

Degradation assays-HPV16 E6 and full-length p53 proteins were cloned in the pXJ40 vector.

Triple pull-downs

HEK-293T cells were grown and maintained in Dulbecco's Modified Eagle's Medium (DMEM), supplemented with 10% FCS and 50 μg/ml of gentamycin at 37 °C with 5% CO 2 and 95% humidity. Cells were seeded in 6-well plates at a concentration of 3×10 5 cells/well. After 24 h, cells were transfected using JetPEI® (Polyplus transfection) with 1 μg pSPICA-N2 plasmid expressing E6, 1 μg of pSPICA-N1 plasmid expressing E6AP protein and 1 μg of BioEase plasmid expressing p53. At 24 h post-transfection, cells were harvested and lysed by freeze-thawing in 100 μl of Renilla Lysis Buffer (Promega, E2820). Cellular lysates were cleared by centrifugation at 13 000 rpm in a microfuge for 15 minutes at 4 °C. Subsequently, 30 μl of pre-equilibrated Streptavidin Mag Sepharose beads (GE Healthcare) were incubated with 60 μl of cellular lysate supernatant for 2 hours at 4°C, thereby allowing capture of biotinylated p53. Streptavidin beads were washed 3 times with TNE buffer (50mM Tris-HCl pH 7.5, 150 mM NaCl, 1% NP40, 1mM EDTA and protease inhibitor) resuspended with 25 μl of protein loading buffer and loaded on a 15% SDS-PAGE gel. E6 and E6AP proteins were detected by Western blotting using rabbit anti-Gluc antibody (NEB, ref. E8023S), whereas p53 was detected using a mouse anti-p53 DO-1 antibody (Life Technologies, ref. 13-4000). The immunoreactive bands were visualized using WesternBright ™ Sirius ™ (Advansta) and LAS 4000 camera. Bands were quantified using ImageGauge software (Fujifilm). Error bars represent standard deviations from three independent experiments.

Cell lines-The HEK-293T cell line was kindly provided by Pierre Charneau (Institut Pasteur, Paris). These cells were authenticated and tested to be mycoplasma free.

p53 degradation assays

E6 and p53 proteins were in vitro translated in the presence of [START_REF] Delano | Benchmarking a luciferase complementation assay for detecting protein complexes[END_REF] S-methionine using the TNT T7 coupled rabbit reticulocyte system. p53 degradation reactions were performed in 10 μl volumes by incubating 2 μl of p53 translation product with either 5 or 2.5 μl of E6 translation products at 28 °C for 2 hours according to previously described protocols [START_REF] Scheffner | The E6 oncoprotein encoded by human papillomavirus types 16 and 18 promotes the degradation of p53[END_REF] .

Reactions were resolved on a 15% SDS-PAGE gel. Gels were exposed to a PhosphorImager screen and scanned using a Typhoon FLA 9500 imaging system (GE Healthcare). Reactive p53 bands were quantified using ImageQuant TL software (GE Healthcare).

p53 degradation activity values reported in Fig. 3 of the main text were derived using the formula (I 0 -I)/I 0 , where I is the intensity of the p53 double band after incubation with E6 and I 0 the p53 signal in the input lane. p53 degradation activity values were normalized to 100% for the reference proteins (E6 4C/4S/wt p53). For p53 mutant proteins (Fig. 3b of maintext), besides the I 0 input control, the I 2h control (corresponding to p53 at time=2h in the absence of E6) has been added to check for the intrinsic stability of p53 mutants. Error bars represent standard deviations from three independent experiments.

Extended Data

Extended Data Fig. 1.

Constructs of the minimal HPV16 E6/E6AP/p53 ternary complex. Green: E6AP residues 403-414 (e6ap 1-12) fused to the C-terminus of MBP (pink) via a AAA linker (red); gold: the HPV16 E6 4C/4S construct [START_REF] Zanier | Solution structure analysis of the HPV16 E6 oncoprotein reveals a self-association mechanism required for E6-mediated degradation of p53[END_REF] (named E6 from here on) comprising four cysteine to serine substitutions (cyan marks) that suppress aggregation mediated by disulfide crossbridging; blue: p53 core domain (p53core).

Extended Data Fig. 2.

Assembly of the E6/MBP-e6ap heterodimer. a, Dynamic light scattering (DLS) analysis of E6/MBP-e6ap samples. Histograms report on the average particles' hydrodynamic radii, whereas error bars indicate size polydispersity. Numbers above the histograms indicate molecular weight estimates assuming a spherical model. This analysis shows that binding to MBP-e6ap enhances the solubility of E6, which, in the unbound state, displays a solubility threshold of 50 μM. However, in the case of E6/MBP-e6ap samples (gray histograms), particle size increases with raising concentration. By contrast, introduction of the F47R mutation in E6 (green histograms) stabilizes particle size to values close to what is expected for a simple heterodimer (≈ 60 kDa). Therefore we conclude that, despite the increase in solubility, E6/MBP-e6ap still undergoes weak self-association via the E6N region hosting Phe47 (see [START_REF] Zanier | Solution structure analysis of the HPV16 E6 oncoprotein reveals a self-association mechanism required for E6-mediated degradation of p53[END_REF] for further discussion). b, Gel filtration analysis of E6/MBP-e6ap samples.

The elution volumes for molecular size markers are reported on top of the figure. The expected elution volumes of a simple E6/MBP-e6ap heterodimer (1x, 60 kDa) and of a dimer of heterodimers (2x, 120 kDa) are indicated. Note the relative small shift in the elution volumes of the different samples as compared to the differences in the hydrodynamic radii (panel a). This suggests that oligomers of the E6/MBP-e6ap heterodimer are rather weak and dissociate on the gel filtration column. See also Supplemental Methods section.

Extended Data Fig. 3.

Interaction of p53core with preformed E6/MBP-e6ap heterodimer. a, Isothermal titration calorimetry (ITC) experiments were performed by titrating increasing amounts of p53core into E6/MBP-e6ap heterodimer samples adjusted to a concentration of 45 μM, which limits heterodimer oligomerization. Note that the F47R mutation in E6 abolishes binding to p53core. See also Extended Data Table 1. b, (Upper panel) Comparison of gel filtration elution profiles of E6/MBP-e6ap/p53core (red dashed line) versus E6/MBP-e6ap samples (black line). Both samples were adjusted to a concentration of 250 μM before loading onto the gel filtration column. The expected elution volumes for p53core (24 kDa), monomeric E6/MBP-e6ap heterodimer (60 kDa) and E6/MBP-e6ap/p53core ternary complex (80 kDa) are indicated. (Lower panel) SDS-PAGE analysis of fractions comprising the elution peak of the ternary complex. Note the significant shift in the elution volumes of the main peak in the two chromatograms, indicating formation of a ternary complex. See also Supplemental Methods section.

Extended Data Fig. 4.

Comparison of the structures of the two E6/MBP-e6ap/p53core heterotrimers (heterotrimers A and B) observed in the asymmetric unit. Green: e6ap fused to the C-terminus of MBP (pink) via the AAA linker (red); gold: HPV16 E6; blue: p53core; spheres: zinc atoms. a, Different orientation of MBP in the two heterotrimers, which results from the different conformations of the two AAA-linkers. b, Superposition of the E6/e6ap/p53 regions of heterotrimers A (green/gold/blue) and B (gray). The backbone rms deviation for the E6/ e6ap/p53 regions of the two heterotrimers was calculated by aligning backbone atoms of residues (i) 12-136 of HPV16 E6, (ii) 371-379 of e6ap and (iii) 109-191 of p53core and found to correspond to 0.9 Å. Regions displaying significant differences are boxed. These regions are the ill-defined C-terminal region of the e6ap peptide (see also Extended Data Fig. 6) and the a4 helix of E6, which is not involved in the E6-p53 interface.

Extended Data Fig. 5.

The structures of the E6/e6ap and p53core subunits of the ternary complex are superimposable with previously solved structures of the E6/e6ap heterodimer and p53core. (Left panel) Superposition of the previously solved E6/e6ap heterodimer (gray) [START_REF] Zanier | Structural basis for hijacking of cellular LxxLL motifs by papillomavirus E6 oncoproteins[END_REF] onto the E6/e6ap subunit of the ternary complex (heterotrimer A, gold/green) determined here. Dashed lines highlight regions of conformational change, namely the N-terminus of HPV16 E6 and the C-terminus of the e6ap peptide. (Right panel) Superposition of previously solved p53core in the unbound state (heterotrimer A, gray) [START_REF] Wang | Structure of the human p53 core domain in the absence of DNA[END_REF] onto the p53core subunit of the ternary complex bound (blue).

Extended Data Fig. 6.

Contributions of the e6ap C-terminal region to ternary complex interface. a, Sequence of wild-type e6ap peptides used in the study. Green: wt e6ap(1-12) corresponding to the peptide used for crystallization of the ternary complex. Gray: wt e6ap(1-16) containing a four amino acid C-terminal extension. b, Electron density (2F O -F C ) map (heterotrimer A) contoured at 1s level for e6ap (green, left panel) and for selected E6 (orange) and p53 (blue) interface residues in the proximity of the C-terminus of e6ap (right panel). Note the lack of electron density data for e6ap residues Glu10, Glu11 and Arg12. Two conformations for the side-chains Glu10 and Arg12 of e6ap and Arg131 of E6 are proposed in the model. c, ITC titration curves showing the interactions of p53core with pre-formed E6/e6ap complexes bearing either point mutations within e6ap C-terminus (E10A, E11A and R12A) or the wt e6ap(1-16) peptide construct. Note that neither the e6ap point mutations nor the e6ap Cterminal boundary affect the interaction between the E6/e6ap heterodimer and p53core. See also Extended Data Table 1.

Extended Data Fig. 7.

Interactions mediated by hrm-HPV conserved residues shaping the conformation of the Nterminal arm of HPV16 E6. Whereas Phe2 contributes to tethering of the N-terminal region to the core of the E6N domain, residues Pro5, Arg8 and Tyr54 are involved in a triple stacking interaction.

Extended Data Fig. 8.

Expression levels of HPV16 E6, E6AP and p53 proteins in for triple pull-down assays. HEK293T were transfected with 1 μg of pSPICA-N1 plasmid (expressing E6AP fused to the C-terminus of the Gaussia princeps luciferase Gluc1 fragment), 1 μg pSPICA-N2 plasmid (expressing the E6 fused to the C-terminus of the Gaussia princeps luciferase Gluc2 fragment) and 1 μg of BioEase plasmid (expressing N-terminally biotinylated p53). Cell lysates were resolved by SDS-PAGE electrophoresis and E6, E6AP and p53 fusion proteins detected by Western blotting. For gel source data, see Supplementary Figure 1.

Extended Data Table 1 Thermodynamic parameters of p53core binding to preformed E6/e6ap complex. Values refer to occupancy-weighted average B-factors (OWAB) (&) The total number of residues in the chain cannot be analyzed: phi and psi angles cannot be analyzed for terminal residues, non-standard residues or incompletely modeled main chain residues. respectively. c, Key interaction networks. Gold fonts: E6 residues; blue fonts: p53 residues; green boxed black fonts: e6ap residues; thick gray dashed lines: non-polar interactions; black lines: polar interactions involving side-chain groups; black dotted lines: polar interactions involving backbone groups; red circles: water molecules. The data (mean ± sd) were obtained from three independent experiments and normalized to 100% for the reference proteins (E6 4C/4S/wt p53). For gel source data see Supplementary Figure 1.

Figure 1 .

 1 Figure 1. Structure of the HPV16 E6/e6ap/p53core ternary complex a, Ribbon (upper panel) and surface (lower panel) representations. Green: e6ap peptide; gold: HPV16 E6; blue: p53core. Spheres: zinc atoms. Boxes indicate sub-interfaces I-III (expanded in Fig.2b). b, Surface representation of E6 colored for residues in atomic contact with p53 (light blue), e6ap (light green), and both p53 and e6ap (dark gray). E6N and E6C: N-and C-terminal zinc-binding domains; HL: helix linker. E6 molecules on the left side of panels a and b are in the same orientation.

Figure 2 .

 2 Figure 2. Intermolecular contacts at the E6-p53 interface a, Alignment of HPV E6 sequences from hrm-and lrm-HPV groups. Histograms: burial of residues at the interface with e6ap (light green) and p53 (light blue). *: positions conserved in hrm-HPV. b, Views of sub-interfaces I-III. Red spheres: water molecules; thick dashed lines: direct polar interactions; thin dotted lines: water-mediated interactions. Upper and lower case fonts: residues mediating polar interactions via side-chain and backbone,

Figure 3 .

 3 Figure 3. Ternary complex assembly and p53 degradation activities of E6 and p53 mutants a, Triple pull-downs. E6AP, E6 and biotinylated p53 were co-expressed in HEK 293T cells. E6AP bound to p53 was detected by Western blotting. All E6 mutations were introduced in the E6 4C/4S construct 12 . *: previously investigated mutants 12 . (Upper panel) Instances of binding efficiencies. I: input; PD: pull-down. (Lower panel) Percentage of E6AP bound to p53. See also Extended Data Fig. 8. b, p53 degradation assays. (Upper panel) Instances of degradation efficiencies showing p53 before and after incubation with 5 or 2.5 μl of E6 translation product. I 0 : p53 band at time=0; I 2h : p53 band at time=2h without E6 (see Methods). (Lower panel) p53 degradation activities with 2.5 μl of E6 product.

Figure 4 .

 4 Figure 4. p53core targeting by HPV E6 a, Complexes of p53core (surface representation) with protein or DNA partners (ribbon representation). Left: p53core bound to HPV E6/e6ap. Right: p53core bound to 53BP2, 53BP1 and BCL-xL cellular proteins, to adenovirus SV LTag oncoprotein and to DNA. p53core binding interfaces: E6 (light orange), other proteins (pink) and DNA (cyan). b, Cartoon summarizing how the LxxLL motif shapes the p53 binding cleft on E6, which recognizes a distinct interface on p53core, enabling targeting of both free and bound p53 molecules. Tetrameric p53 is shown with one subunit colored for interfaces binding to HPV E6 (yellow), DNA (cyan and purple), and other host proteins (pink and purple).

Table 2

 2 Data collection and refinement statistics.

	e6ap	a N	b c min /c max	K D (μM)	ΔH (kcal/mol) TΔS (kcal/mol) ΔG (kcal/mol)
	wt c6ap(1-12)	0.826±0.016	1.8/2.4	22.1±4.5	-17.4±1.5	-0.926±0.140	-16.5
	wt e6ap(1-16)	0.516±0.046	1.8/2.7	20.5±3.9	-19.1±2.7	-1.100±0.299	-18.0
	E10A e6ap(1-12) 0.827±0.080	3.1/3.3	14.1±0.6	-17.1±0.5	-0.878±0.051	-16.2
	E11A e6ap(1-12) 0.679±0.146	2.6/4.0	14.3±4.6	-13.8±0.1	-0.599±0.012	-13.2
	R12A e6ap(1-12) 0.719±0.111	1.0/2.2	34.6±13.6	-13.9±2.7	-0.651±0.250	-13.2
	All ITC experiments were performed at 25 °C.
	a N refers to the molar ratio of p53core : E6/e6ap complex.
	b c value is defined as: c=n[E6/e6ap]/K D .
			E6/e6ap/p53
	Data collection
	Space group		P 2 1
	Cell dimensions
	a, b, c (A)		78.15 129.41 82.26
	α, β, γ (°)		90, 92.4, 90
	Resolution (Å)		50-2.25 (2.33-2.25) (*)
	R sym or R merge		6.0 (68.4)
	I/σI		17.6 (1.8)
	Completeness (%)		99.5 (99.6)
	Redundancy		3.2 (3.1)
	Refinement
	Resolution (Å)		50-2.25
	No. reflections		77282
	R work /R free (%)		19.4/24.6 (20.9/33.0)
	No. atoms		12034
	Protein		11593
	Ligand/ion		84
	Water		354
	B-factors (Å 2 ) (#)
	Protein		50
	MBP-e6ap (chains A, B) 50, 58
	p53 (chains C, D)	43, 45

Extended Data

Nature. Author manuscript; available in PMC 2016 July 20.

Acknowledgments

This work received institutional support from CNRS, Université de Strasbourg, INSERM and Région Alsace. The work was supported by grants from 'Ligue contre le Cancer', NIH (grant R01CA134737 to SVP), ANR (ANR-13-JSV8-004-01), Instruct (ESFRI), the French Infrastructure for Integrated Structural Biology (FRISBI) and 'Fondation pour La Recherche Medicale' (fellowship to FXRF). The authors thank Pierre Poussin-Courmontagne (Structural Biology and Genomics platform, IGBMC, Strasbourg), Eric Ennifar (IBMC, Strasbourg), Vincent Olieric (SLS, Villigen) and Bruno Kieffer (IGBMC, Strasbourg) for advice.

Supplementary Material

Refer to Web version on PubMed Central for supplementary material.


domain: stat math spi.auto

text:
Student Member, IEEE, Mustapha Benjillali, Senior Member, IEEE, Samir Saoudi, Senior Member, IEEE Yassine Hmamouche

Senior Member, IEEE Daniel Benevides Da Costa

Uplink Energy Efficiency Distribution with Aerial Users in Cellular Networks

Keywords: Power Control, Signal-to-Interference Ratio, Stochastic Geometry, Unmanned Aerial Vehicles (UAVs)

The power consumption of future user equipments (UEs) will be affected by the projected growth in their computing capacity, while data throughput may be affected by emerging aerial UEs with specific radio propagation conditions compared to terrestrial UEs. In such a context, this letter evaluates a key metric of interest, namely the probability that the uplink energy efficiency (EE) at a typical ground base station will be higher than a predefined threshold. We first characterize the priority bias of each UE layer as a function of long-term shadowing and systemlevel parameters to assess its penetration rate, i.e., the amount of active UEs from each tier among the total population of active UEs. Next, tractable approximations of the desired signal and the interference distribution are performed, enabling to derive the uplink EE. Our results demonstrate that an aggregation of the system-level parameters through the aerial priority bias needs to meet a given constraint to mitigate interference from aerial UEs and enhance the uplink EE of ground UEs. Monte-Carlo simulations validate the accuracy of our analytical results.

I. INTRODUCTION

Given their agility and flexible deployment, interest in unmanned aerial vehicles (UAVs) technology is rapidly growing, opening doors to various realms of application. The ongoing technological advances and upcoming generations of wireless networks such as fifth generation (5G) and beyond (B5G), will enable these equipments to be enhanced with many new sensors and seamless connectivity, making them more robust and more useful than their older versions.

Despite their expected benefits, UAVs as aerial user equipment (UE) in communication networks, can nevertheless have a detrimental impact on the performance of terrestrial UEs, which are often assigned more critical tasks than UAVs (e.g., monetary transactions, health-care services) [START_REF] Van Der Bergh | LTE in the sky: trading off propagation benefits with interference costs for aerial nodes[END_REF]. The Third Generation Partnership Project (3GPP) has involved in Release 15 a technical study to assess the capability to serve aerial UEs through Long Term Evolution (LTE) deployments where base station (BS) antennas are targeting terrestrial UEs [START_REF]Study on enhanced LTE support for aerial vehicles[END_REF]. It is particularly observed from simulations that an increased density 1 of aerial UEs will significantly increase the uplink Y. Hmamouche and S. Saoudi are with the Department of Signal and Communications, IMT-Atlantique, Brest 29200, France (e-mails: {yassine.hmamouche, samir.saoudi}@imt-atlantique.fr). M. Benjillali is with the Communication Systems Department, INPT, Rabat 10100, Morocco (e-mail: benjillali@ieee.org). D. B. da Costa is with the Department of Computer Engineering, Federal University of Ceará, Sobral 62010-560, Brazil (e-mail: danielbcosta@ieee.org). This work was supported by Pôle de Recherche Avancée en Communications (PRACOM) and the Regional Council of Brittany, France. 1 In this work, we consider an aggregated measure more general than aerial UE density, namely the aerial priority bias that we will discuss later.

interference on ground BSs. This is due to the observation that a typical aerial UE experiences line-of-sight propagation conditions with a higher probability to more cells as compared to a typical terrestrial UE. This increase in uplink interference would require a higher resource utilization level to maintain a similar level of throughput for ground UEs. On the other hand, increasing the resource utilization level further magnifies the uplink interference in the network, and hence further degrades the uplink throughput of both aerial and terrestrial UEs.

Considering deterministic locations and fixed number of ground BSs, terrestrial UEs, and aerial UEs, most research efforts have generally leveraged field measurements and simulations [START_REF] Amorim | Measured uplink interference caused by aerial vehicles in LTE cellular networks[END_REF], algorithmic analysis [START_REF] Mei | Cellular-connected UAV: uplink association, power control and interference coordination[END_REF], and optimization theory [START_REF] Liu | Multi-beam UAV communication in cellular uplink: cooperative interference cancellation and sum-rate maximization[END_REF] to evaluate the impact of aerial UEs on the performance of terrestrial LTE cellular networks. The above techniques are typically time-consuming, require customized setups for each experiment, and need complex and efficient algorithms. Hence, increasing need for tractable analytical models. To our knowledge, there is no analytical model available in the literature to capture the impact of UAVs system-level parameters (density, height, consumed power, and power control factor) on uplink performance metrics of cellular networks.

Stochastic geometry and its inherent point process theory is considered as a powerful mathematical tool for the systemlevel analysis of wireless networks [START_REF] Andrews | A tractable approach to coverage and rate in cellular networks[END_REF]. However, uplink analysis is highly challenging due to the coupling in active UEs locations following the use of orthogonal multiple access schemes such as orthogonal frequency-division multiple access (OFDMA), and also due to distance-dependent power control schemes inducing dependency between BSs and UEs locations. Several generative models have been developed in the literature to assess the uplink performance in terrestrial single-tier cellular networks [START_REF] Novlan | Analytical modeling of uplink cellular networks[END_REF], multi-tier wireless networks [START_REF] Singh | Uplink rate distribution in heterogeneous cellular networks with power control and load balancing[END_REF], and dense cellular networks [START_REF] Ding | Uplink performance analysis of dense cellular networks With LoS and NLoS transmissions[END_REF]. Despite these analytical models, there are still important unexplored leads that need to be addressed. For instance, i) considering heterogeneity at the UE level (terrestrial and aerial UEs) as opposed to BS level (small cells, macro cells). The uplink analysis under the latter is revealed to be statistically equivalent to that under the single tier setup [START_REF] Singh | Uplink rate distribution in heterogeneous cellular networks with power control and load balancing[END_REF]Corollary 4]. Also, ii) previous works have only addressed typical performance metrics such as coverage probability and throughput. Adjusting systemlevel parameters to improve these metrics would however have a detrimental impact on other important metrics such as the power consumption at UEs interested in having longer battery autonomy. Our contributions can be summarized as follows:

• We consider two classes of terrestrial and aerial UEs with distinctive parameters in terms of long-term shadowing, and system-level parameters, and introduce a measure of priority between UE tiers, namely the UE priority bias.

The rationale is to evaluate the process of active uplink UEs from each tier and then derive the distribution of the serving UE in uplink. • Deriving the process of interfering UEs is quite challenging in the uplink analysis. In our setup, we approximate it with an inhomogeneous Poisson point process (PPP) over an exclusion region defined by the tradeoff between interfering UEs and the serving UE average received power at the typical BS.

• Finally, we derive the distribution of the uplink energy efficiency (EE), enabling us to evaluate the amount of UEs with a good tradeoff between throughput and power consumption. Next, we illustrate the detrimental effect of increasing aerial priority bias on the EE. An analytical constraint based on the Lambert W function and mapping main system-level parameters is identified for proper operational regime of cellular networks with aerial UEs. Notations: P (.) and E (.) stand for probability and expectation measures. L X (s) = E e -sX is the Laplace transform of a random variable X evaluated at s. For a ∈ R, Γ(a) = ∞ 0 t a-1 e -t dt is the Gamma function. W 0 is the principal branch of the Lambert W function defined by the lagrange inversion theorem as

W 0 (x) = n≥1 (-n) n-1 n! x n .
II. SYSTEM MODEL We consider the uplink of a cellular network made of terrestrial BSs modeled according to a 2D homogeneous PPP Ψ b {y i } with density λ b in the Euclidean plane P ⊂ R 2 , and a vertical set of heterogeneous UEs (having data to transmit in uplink), comprising two classes of UEs; terrestrial UEs modeled according to a HPPP Ψ t with density λ t , and aerial UEs deployed at an average altitude h a such that their projection in P is modeled according to a homogeneous PPP Ψ a with density λ a . Without loss of generality, the typical BS y 0 at the origin O, is the object of the analysis.

We consider standard power-law path-loss between y 0 and UEs from Ψ t and Ψ a as ℓ t (r) = Kr -α and ℓ a (r

) = K h 2 a + r 2 -α
2 , respectively, where r is the horizontal distance between the UE of interest and y 0 , α > 2 is the pathloss exponent, and K = 3.10 8 4πf 2 is the free-space pathloss such that f is the transmission frequency. Multipath fading of the link between y 0 and a UE x, is incorporated by a positive independent and identically distributed (i.i.d.) Rayleigh random variable (RV) g x with unit average power, i.e., g x ∼ exp(1). Also, for i ∈ {t, a}, the link between y 0 and a UE from Ψ i is subject to shadowing such that its power is modeled by log-normal RVs χ i with mean µ i (in dB) and standard deviation σ i (in dB). Based on the displacement theorem, shadowing effect can be absorbed into the HPPP density. In this way, the displaced HPPP is still noted Ψ i with density

λ i = λ i E χ 2/α i < ∞, where E χ 2/α i = exp ln(10) 5 µi α + ln(10) 5 √ 2 σi α 2 .
The total power usage for the uplink transmission of a UE from Ψ i is P i usage = P s,i + P d,i , where P s,i and P d,i

A t = 1 -exp -πλ t δ 2 + 2πλ t e πλah 2 a × ∞ δ r exp -π λ t r 2 +λ a K ǫt-ǫa P a P t 2 α(1-ǫa ) r 2(1-ǫ t ) 1-ǫa dr, (1)
A a = 2πλ a × ∞ 0 r exp -π λ a r 2 +λ t K ǫa-ǫt P t P a 2 α(1-ǫ t ) r 2 +h 2 a 1-ǫa 1-ǫ t dr, ( 2
)
where δ = h 1-ǫa 1-ǫ t a K ǫa-ǫt Pt Pa 1 α(1-ǫ t ) .
Proof. For i ∈ {t, a}, the complementary cumulative distribution function (CCDF) of r i , the smallest distance to contact a UE from Ψ i , is obtained from the null probability of the PPP Ψ i as

F ri (u) = P (r i > u) = exp -πλ i u 2 .
(

) 3
From the abstraction of A t in Assumption 1, we get

At = P K 1-ǫt Ptrt -α(1-ǫt) ≥ K 1-ǫa Pa ra 2 + h 2 a - α(1-ǫa ) 2 (4) = E rt P ra 2 ≥ K ǫt-ǫa Pa Pt 2 α(1-ǫa ) u 2(1-ǫ t ) 1-ǫa -h 2 a rt = u (5) = - ∞ 0 F ra   max 0, K ǫt-ǫa Pa Pt 2 α(1-ǫa ) u 2(1-ǫ t ) 1-ǫa -h 2 a   dF rt (u) . (6)
Exact expression of A t is then obtained by using CCDFs in (3) and splitting the interval of integration based on δ. A similar approach is followed to derive A a .

Special cases: when ǫ t = ǫ a = ǫ, Lemma 1 can be simplified under a closed-form expression as

A a = 1 -A t (7) = λ a P 2 α(1-ǫ) a λ t P 2 α(1-ǫ) t + λ a P 2 α(1-ǫ) a exp -πλ t h 2 a P t P a 2 α(1-ǫ) . (8)
If we also have P a = P t , (8) can be further simplified as

A a = 1 -A t = λ a λ t + λ a exp -πλ t h 2 a . (9)
For i ∈ {t, a}, we denote by r i x the horizontal distance between a given UE x ∈ Ψ i and its nearest BS. Particularly, r t and r a are, respectively, the horizontal distance from y 0 to its nearest UE from Ψ t and Ψ a . Also, we denote by S the serving tier, i.e., S = Ψ t or S = Ψ a , and by r 0 the horizontal distance from y 0 to its serving UE x 0 ∈ S. The distance from y 0 to an interfering UE x from Ψ i \ {x 0 } is denoted by d i

x . Fig. 1 illustrates the setup of our system model.

For the sake of simplicity, we ignore thermal noise and focus our analysis on the interference-limited regime [START_REF] Novlan | Analytical modeling of uplink cellular networks[END_REF], [START_REF] Singh | Uplink rate distribution in heterogeneous cellular networks with power control and load balancing[END_REF]. In such a context, the signal-to-interference ratio (SIR) at y 0 is given by

SIR(x 0 ∈ Ψ i ; y 0 ) = g x0 P i (ℓ i (r 0 )) 1-ǫi I t x0,i + I a x0,i , (10)
where for i, j ∈ {t, a}, I j x0,i is the interference generated by UEs from Ψ j \ {x 0 } conditioned on a serving UE x 0 ∈ Ψ i . It is expressed as

I j x0,i = x∈ Ψj \{x0∈ Ψi} g x P j ℓ j (d j x ) ℓ j (r j x ) -ǫj . (11)
III. UPLINK ENERGY EFFICIENCY DISTRIBUTION In this section, we provide the key analytical framework to derive the uplink EE distribution under a setup with terrestrial and aerial UEs.

Definition 1. The uplink rate (in bps) at the typical BS y 0 when it is served by

x 0 ∈ Ψ i (i ∈ {t, a}) is R i = B N + 1 log 2 1 + SIR(x 0 ∈ Ψ i ; y 0 ) , ( 12
)
where B is the total effective uplink bandwidth in Hz and N is the total number of UEs from Ψ t ∪ Ψ a \ {x 0 ∈ Ψ i } with orthogonal projection inside C 0 .

Definition 2. (Energy efficiency coverage).

EE coverage E c is defined as the probability that the uplink EE measured at the typical BS is higher than a predefined threshold T. Formally,

E c = i∈{t,a} P R i P s,i + P d,i ≥ T, S = Ψ i . ( 13
)
The EE distribution is completely characterized by the EE coverage. Also, N is a RV depending on the serving area of the typical BS y 0 , i.e., C 0 , and the priority bias that governs the number of active terrestrial and aerial UEs in the uplink. Assumption 3. Given the orthogonal allocation of channel resources, it is reasonable to consider that the number N of competing UEs is uncorrelated with SIR at y 0 .

In this way, the EE coverage can be derived as

Ec= i∈{t,a} P   log 2 1 + SIR(x0 ∈ Ψi; y0) Ps,i + Pd,i ≥ T (N + 1) B , S = Ψi   (14) (a)
= m≥0 i∈{t,a}

P (N = m) P gx 0 ≥ si (m, r0) I t x 0 ,i + I a x 0 ,i , S = Ψi ( 15
) (b) = m≥0 i∈{t,a} P (N = m) Er 0 L I t x 0 ,i si (m, r0) L I a x 0 ,i si (m, r0) , (16)
where (a) comes from the independence between the distribution of the number of competing UEs to x 0 and the received SIR from x 0 , (b) comes from g x ∼ exp(1) and the definition of the Laplace functional, where for i ∈ {t, a},

s i (m, r 0 ) = 2 T(m+1) B ( P s,i +P i (ℓ i (r 0 )) -ǫ i ) -1 Pi(ℓi(r0)) 1-ǫ i
, and the probability mass function (PMF) of N is [START_REF] Dhillon | Downlink rate distribution in heterogeneous cellular networks under generalized cell selection[END_REF] P (N = m) ≃ 3.5 .

(

) 17
We derive next the supplementary terms for the computation of the uplink EE coverage.

A. Distribution of the serving UE

After associating one UE per each BS, the probability density function (PDF) of the horizontal distance between the typical BS y 0 and its serving UE x 0 from Ψ t is derived as

f r0 (u, S = Ψ t ) = f rt (u) × P K -ǫt P t u -α(1-ǫt) > K -ǫa P a r 2 a + h 2 a -α(1-ǫa ) 2 (18) = 2π λ t u exp -π λ t u 2 × 1 , u < δ exp -π λa K ǫt-ǫa Pa Pt 2 α(1-ǫa ) u 2(1-ǫ t ) 1-ǫa -h 2 a , otherwise. ( 19
)
Similarly, the PDF of the horizontal distance between the typical BS y 0 and its serving UE x 0 from Ψ a is

f r0 (u, S = Ψ a ) = f ra (u) × P K -ǫa P a u 2 + h 2 a -α(1-ǫa ) 2 > K -ǫt P t r -α(1-ǫt) t (20) = 2π λ a u exp -π λ a u 2 × exp -π λ t K ǫa-ǫt P t P a 2 α(1-ǫ t ) u 2 + h 2 a 1-ǫa 1-ǫ t . (21)
The above expressions of the serving distance PDF are different from those defined in [START_REF] Novlan | Analytical modeling of uplink cellular networks[END_REF]- [START_REF] Ding | Uplink performance analysis of dense cellular networks With LoS and NLoS transmissions[END_REF], where only one homogeneous class of UEs is considered.

Approximating the process of interfering UEs is quite challenging due to coupling in UEs locations. For tractability, we adopt a similar abstraction as in [START_REF] Singh | Uplink rate distribution in heterogeneous cellular networks with power control and load balancing[END_REF], [START_REF] Andrews | A primer on cellular network analysis using stochastic geometry[END_REF], where for i ∈ {t, a}, Assumption 4. The process of interferers from Ψ i , conditioned on the serving UE, is modeled by an inhomogeneous PPP of density λ i 1 -exp -π λ i v 2 , where v is the horizontal distance to a given interferer. Assumption 5. Conditioned on an interfering UE x ∈ Ψ i , the PDF of r i

x is expresssed under a truncated version as

f r i x w|d i x = v = 2π λ i w exp -π λ i w 2 1 -exp -π λ i v 2 , ∀w ≤ v. ( 22
)
The exclusion region of interfering UEs under our setup is defined by the observation that the average received power from any interfering UE at y 0 needs to be lower than that received from the serving UE x 0 . Otherwise, the interfering UE will be associated to the typical BS y 0 . Formally,

P i ℓ i (d i x ) ℓ i (r i x ) -ǫi ≤ P j (ℓ j (r 0 )) 1-ǫj , ∀i, j ∈ {t, a} . (23)

B. Interference from active terrestrial users

For j ∈ {t, a}, the Laplace functional of the interference generated by UEs from Ψ t when the serving UE belongs to Ψ j , is obtained as

L I t x 0 ,j (s) = E    exp   -s x∈ Ψt\{x 0 ∈ Ψ j } gxPtK 1-ǫt d t x -α r t x αǫt       (24) = E    x∈ Ψt\{x 0 ∈ Ψ j } E r t x 1 1 + sPtK 1-ǫt (d t x ) -α (r t x ) αǫt    (25) (a) = exp -4π 2 λ 2 t ∞ 0 v 0 vwe -π λtw 2 1 + K ǫ t -1 sPt v α w αǫ t dwdv , (26)
where (a) is by averaging first over the PDF of r t x in Assumption 5. Next, we use the probability generating functional (PGFL) theorem where the density of the interference field is defined in Assumption 4. The integration range of d x = v starts from 0, since the closest interfering UE can be closer to y 0 than the desired UE x 0 . This is possible from (23) since, for an interferer UE from Ψ t with a distance to its closest BS of r t x = r 0 /n (n > 1), it can be closer to y 0 by r 0 /n ǫt .

C. Interference from active aerial users

The Laplace functional of the interference generated by UEs from Ψ a when the serving UE belongs to Ψ t and distant from y 0 by r 0 = u, is obtained as

L I a x 0 ,t (s) = exp      -4π 2 λ 2 a ∞ ̟(u) v 0 vwe -π λaw 2 1 + K ǫa -1 sPa (v 2 +h 2 a ) α 2 (w 2 +h 2 a ) αǫa 2 dwdv      , (27)
where the starting point of the integration range with respect to d x = v is derived from (23) as

̟ (u) =    0 , u < δ ha Pa Pt K ǫt-ǫa 2 α u 2(1-ǫ t ) h 2(1-ǫa ) a -1 , otherwise. (28)
Similarly, the Laplace functional of the interference generated by UEs from Ψ a when the serving UE belongs to Ψ a and distant from y 0 by r 0 = u, is obtained as

LIa x 0 ,a (s) = exp      -4π 2 λ 2 a ∞ π(u) v 0 vwe -π λaw 2 1 + K ǫa -1 sPa (v 2 +h 2 a ) α 2 (w 2 +h 2 a ) αǫa 2 dwdv      , (29)
where

π (u) = h a u 2 h 2 a + 1 1-ǫa -1.

IV. NUMERICAL RESULTS

We consider a typical cellular network where each BS has an average of 80 smartphones and 20 quadcopter UAVs within its coverage area, i.e., λ t = 80 λ b and λ a = 20 λ b . We have the following standard parameters: α = 4, f = 2.1 GHz, B = 5 MHz, µ t = µ a = 0 dB, P t = 33 dBm, P a = 36 dBm, P s,t = 0.5 Watts, and P s,a = 2.5 Watts.

We first need to validate the expression of priority bias derived in Lemma 1. Fig. 2 illustrating the simulated and analytical expression of the aerial priority bias shows that the analytical expression in Lemma 1 is perfectly accurate. In Fig. 3, we validate the analytical expression of the EE coverage under the mean load setup, i.e., P N = 1 + 1.28 λt+λa λb ≃ 1 [START_REF] Dhillon | Downlink rate distribution in heterogeneous cellular networks under generalized cell selection[END_REF]. It has been shown that the EE coverage decays when lowering UAVs height or increasing their density by making aerial shadowing more variable, i.e., increasing σ a . This is a similar result to [START_REF] Dhillon | Downlink rate distribution in heterogeneous cellular networks under generalized cell selection[END_REF], where shadowing is revealed to be a natural load balancing bias between tiers of BSs in the downlink. Here, it plays the role of a natural priority balancing between several classes of uplink UEs.

In Fig. 4, we illustrate the mean load terrestrial EE coverage, i.e., E c S = Ψ t , and the aerial priority bias A a as a function of the power control factor ǫ = ǫ t = ǫ a and UAVs height h a . E c S = Ψ t is typically monotonically decreasing with the aerial priority bias. Reducing the latter can therefore reduce the detrimental effect of UAVs on the EE of terrestrial UEs.

To achieve this, we can act not only by reducing UAVs density (as mentioned in [START_REF]Study on enhanced LTE support for aerial vehicles[END_REF]), but also by acting on other system parameters such as altitude and the power control factor (see Fig. 2). In the general case, assuming a properly operating LTE cellular network for A a ≤ β < 1, we get from (8) (31)

V. CONCLUSION This paper proposed a tractable analytical framework to derive the uplink EE coverage under a setup where two classes of terrestrial and aerial UEs are considered. It has been particularly shown that increased tendency of terrestrial BSs to be connected to aerial UEs, i.e., increased aerial priority bias (not only UAVs density as reported in [START_REF]Study on enhanced LTE support for aerial vehicles[END_REF]), will have a detrimental effect on the uplink EE of ground UEs. Assuming similar power control factors for terrestrial and aerial UEs, we have identified a compact analytical constraint for a proper operational regime of the network in terms of EE coverage.

Future generalization of this work will consider constrained transmit power at the level of each UE class, in addition to multi-tier BSs. The rationale is to investigate the impact of a decoupled uplink and downlink association scheme on EE.

Fig. 2 .hhFig. 4 .

 24 Fig. 2. Aerial priority bias as a function of terrestrial and aerial power control factors and UAVs altitude, when λ b = 10 -2 , σt = 4 dB, and σa = 8 dB.

  is the inverse relation of f (w) = we w , we obtain from (30) a transcendental constraint as