Yanis Meziani

Besma Khalfoun

Sara Bouchenak

Sonia Ben Mokhtar

Vlad Nitu

Enforcing Adaptive Location Privacy with Federated Learning

Keywords: Mobility Data, Location Privacy, Re-identification Attacks, Federated learning. 1

Preserving the privacy of mobility data has been the center of active research in the last decade as this data may reveal sensitive information about individuals (e.g., home, work places, political, religious, sexual preferences). In this context, a large variety of Location Privacy Protection Mechanisms (LPPMs) have been proposed. To employ LPPMs more effectively for the benefits of the users' privacy, adaptive solutions that dynamically combine LPPMs have also been investigated. These solutions apply various LPPMs on a given trace and choose the one that better meets privacy and utility requirements. To meet this objective, adaptive solutions often rely on a trusted proxy that gathers users' traces and apply LPPMs locally. In this paper we release this assumption by designing an approach, i.e., EDEN, where mobility data never leaves the user's device before being protected by an appropriate LPPM. Experimental evaluation performed on real mobility traces shows that EDEN with its local and adaptive strategy outperforms individual LPPMs both in terms of privacy and utility metrics.

Introduction

The last decade has witnessed the apparition of a wide variety of location based services (LBSs), which are extensively used by increasing numbers of users. While LBSs offer useful services to the customers (e.g., finding points of interests, geo-gaming), the latter massively collect (and possibly share/sell) user location data. For instance, recent studies 1 have shown that among 30000 sampled iOS applications available in the App store, 51.1% collect location data when the application is in use. The downside is that collecting users' location data opens many privacy threats if the latter falls between the hands of curious/malicious adversaries. For instance, mobility data may very well reveal a user's home and workplace, health status or even religious or sexual preferences if the latter regularly visits health centers, worship places or libertine places (respectively). In order to mitigate the above threats, the research community has been actively proposing Location Privacy Protection Mechanisms (LPPMs). Examples of proposed LPPMs include Geoindistinguishability [START_REF] Andrés | Geoindistinguishability : Differential privacy for location-based systems[END_REF], which adds spacial noise to a user's GPS coordinates, Promesse [START_REF] Primault | Time distortion anonymization for the publication of mobility data with high utility[END_REF], which removes places where the user stops and Trilateration [START_REF] Huang | Search locations safely and accurately : A location privacy protection algorithm with accurate service[END_REF], which generates dummy locations to obfuscate the user's real location. In this context, one of the issues that has been exhibited by the research community is that there is no one size fits all LPPM that effectively protects users' location data in all contexts. An additional dimension that makes it more complex to protect users' mobility data is that each LPPM can be configured differently hence offering a different privacy vs utility trade-off. To deal with this issue, solutions that try to find the best LPPM (and its corresponding proper configuration) to protect a given user mobility data at a given point in time have been devised (e.g., MooD [START_REF] Khalfoun | Mood : Mobility data privacy as orphan disease : Experimentation and deployment paper[END_REF]). These solutions usually rely on known attacks (e.g., [START_REF] Gambs | Show me how you move and i will tell you who you are[END_REF][START_REF] Maouche | Ap-attack : a novel user re-identification attack on mobility datasets[END_REF]) and utility metrics (e.g., the spacial distortion induced by the LPPM compared to raw traces) to assess the effectiveness of LPPMs. Specifically, given a mobility trace on which a set of LPPMs are applied, adaptive solutions would choose the one that better resists state-of-the-art attacks while preserving data utility above a given threshold. However, these solutions often rely on the assumption that a trusted proxy server collects users' mobility data, runs the attacks and consequently apply the appropriate LPPM. In this paper we aim at releasing this assumption by designing an approach where mobility data never leaves the user's device before being protected by an appropriate LPPM. Specifically, we present EDEN a solution that relies on federated learning to train a decentralized attack model called FURIA. This attack is then used locally along with a utility metric in order to choose the most appropriate LPPM for a given mobility trace. We evaluated EDEN with a set of experiments using real mobility data. We used EDEN to choose the best LPPM and its corresponding configuration among nine possibilities (3 LPPMs and 3 different configurations). The results show that EDEN outperforms individual LPPMs both in terms of privacy (better resilience against FURIA our decetralized attack) and in terms of data utility. The remaining of this paper is structured as follows. First, we present in Section 2, Background and related work. Then, we describe EDEN design principles in section 3. Further, in section 4, we proceed to an experimental evaluation of our attack. Finally, we conclude in section 5.

Background and Related Work

There are two parties involved in the context of our work : the Location-Based Service (LBS) represented by an LBS server and the users represented by their mobile devices. We suppose that users send queries to the LBS server with raw mobility data (single or multiple mobility records). Therefore, the LBS provider has access to all raw mobility data of all users. We consider the LBS as an honest but curious entity. It provides the users with the requested information according to their locations but it may store the collected raw mobility data and exploit it malliciously. The stored mobility traces are noted as the background knowledge of the LBS, BK = {t 1 , t 2 , . . . , t m }. Where t i corresponds to the mobility trace of user u i . After that, users start using Location Privacy Protection Mechanisms (LPPMs) in order to protect their mobility data. As of now, only obfuscated mobility data are sent to the LBS. The latter tries to re-identify the owner of the data based on BK. In the literature, we have many re-identification attacks suggestion. Among them, we find the POI-attack [START_REF] Primault | Differentially private location privacy in practice[END_REF], which uses the Point Of Interests (POIs) to represent mobility of a user, POIs are locations where users spent an amount of time. PIT-attack [START_REF] Gambs | De-anonymization attack on geolocated data[END_REF] also uses Mobility Markov chains where states are POIs and edges represent the probability of transition between POIs. AP-attack [START_REF] Maouche | Ap-attack : a novel user re-identification attack on mobility datasets[END_REF] describes user mobility as a heatmap which inspires our work.

EDEN Design Principles

System Model

Let U = {U 1 , U 2 , . . . , U n } be the set of users. Each user U i holds a mobile device which collects and stores her raw mobility data. These data are used by geo-located services in order Compas'2020 : Parallélisme/ Architecture/ Système / Temps Réel MILC -Lyon, France, du 30 juin au 3 juillet 2020 to respond to user's requests. A user request contains the user ID (e.g., IP address), the user location (i.e., GPS latitude and longitude), the time at which the request is issued, and the actual requested method (e.g., best restaurants nearby). We consider the record r j = (lat j , lng j , t j ) as a mobility data record where lat j and lng j respectively correspond to the latitude and the longitude of GPS coordinates, and t j is a timestamp. A sequence of mobility data records forms a mobility trace T j = {r 1 = (lat 1 , lng 1 , t 1 ), r 2 = (lat 2 , lng 2 , t 2 ), . . .}.

Overview of EDEN

EDEN (Enforcing aDaptive location privacy with federated lEarNing) is a user side mobility data protection system. Thus, user sensitive data are kept locally and are not shared with other entities, which increases privacy guarantees. EDEN's main objective is to protect users mobility data in online/semi-online services. EDEN includes and uses a distributed user reidentification attack called FURIA (Federated User Re-Identification Attack). This allows EDEN to better assess the resilience of protected data to privacy attacks. The overall execution process of EDEN and FURIA is as follows. During the day, all or a subset of users run mobile applications and get access to the LBS services at any time of the day. The mobility data received by the LBS provider is continuously protected by EDEN thanks to FURIA which improves its users mobility knowledge every day. For a given day j, let T i = {T i 1 , T i 2 , . . . T i k } be the set of mobility traces of user U i collected over the whole day. At the end of the day (e.g., at midnight), each user gets back the global model of FURIA ,transforms her set of mobility traces T i into a vector of features, trains the model locally and sends it back to FURIA maintainer. Once The FURIA maintainer gets the users updates, it aggregates the gradients and produces a updated global model which continuously learns new discriminative mobility patterns. The latter helps to uniquely identify users. User mobility data features are extracted from raw mobility traces in order to represent spatio-temporal behavior of users in a compact way. The following mobility features are considered : Spatial feature. The spatial information (i.e. lat and lng) is projected on a heat map. A heat map is a set of cells of equal size. For each cell, the proportion of mobility records in a given trace T j that belong to that cell is computed. This corresponds to the cell visit rate.

Temporal feature. The temporal information is considered to differentiate similar mobility patterns between day-night shifts, e.g., a user living near the working place of another user. If temporal information is not taken into account, they may have similar heat maps but at different times of the day. A simple, yet effective, temporal information that is the average time of the day of all the records in a given T j . This is convenient in the case of online/semi-online LBS, where mobility traces are minutes length (e.g., online session-based services) or hours length (e.g., crowd-sensing systems), without exceeding a day. Additional features. Other types of information are extracted to enrich the user mobility profile. For instance, the number of mobility data records available in the trace T j is considered. This allows to represent location-based service usage intensity. We also extract the centroid of the mobility trace T j (i.e., centroid's latitude and centroid's longitude), to capture the average position of the user in the map.

EDEN Architecture

The architecture of EDEN is depicted in the left part of Figure 1. EDEN takes as input a user mobility trace T and a set of LPPMs with various configuration (i.e. low, medium and high impact on mobility data) and returns as output an obfuscated mobility trace T ′ i which will be publicly released for crowd-sensing campaign or sent to LBS server for a particular purpose (e.g., search for nearby market). It has four main components, the first component Apply LPPMs aims at applying the considered LPPMs on the raw mobility data T stored on the user Compas'2020 : Parallélisme/ Architecture/ Système / Temps Réel MILC -Lyon, France, du 30 juin au 3 juillet 2020 mobile device (step 1 ). The second component Format Data transforms the different obfuscated versions of the raw mobility trace, i.e., {T ′ 1 , T ′ 2 , . . . , T ′ n} into feature vectors as enumerated in section 3.1, (step 2 ). the third component Global Model FURIA is the crucial component in the system. Once a day, the user retrieves the latest version of the Global Model FURIA. The latter evaluates the re-identification of each vector of features of the obfuscated data, (step 3 ). If the attack succeeds in predicting the right identity label associated to the mobility trace, the latter will be deleted and not shared outside the user device. Otherwise, the protected mobility trace is potentially elected to be sent to the LBS server. Finally, the last component Best coverage, only the protected mobility trace against the most up to date joint model of FURIA with the maximal area coverage is retained. Note that the area coverage is computed between the original and the obfuscated mobility trace, T and T ′ i respectively, to measure the covered cells by T ′ i in total cells of T [13], (step 4 ). Once the protected mobility trace with best utility is produced and sent to the LBS server, the latter treats the user's request and returns back the response to the user's device, (step 5 ).

L1(T)=T'1

Protected <lat2,lng2,t2> <lat3,lng3,t3> <latn,lngn,tn>

L2(T)=T'2 Ln(T)=T'n

FURIA Architecture

FURIAS's global model is an essential part of EDEN. It applies federated machine learning over decentralized user mobility data, in order to learn user mobility models from different users, in a privacy preserving way. It involves the following parties, c.f., Figure 1 -right side : (i) mobile user devices where raw mobility data are kept at user's own device, (ii) a master server responsible of the Federated modeling and processing of the user re-identification attack, i.e., an attack which aims to associate an anonymous mobility trace to its originating user identity. First, The Master server initializes randomly a straightforward classification algorithm which is Logistic regression and sends it to all participants (step 1 and 2 ). This model is denoted as FURIA 0 . Each participant U j transforms its raw mobility data of the day to feature vectors (step 3 ) and trains the model FURIA 0 locally on the generated vectors. Notice that the training phase is done in the user device and not in the Master server. Once all participants have finished the first learning round, they send their local updates (i.e. gradients) of their current local models to the Master server, (step 4 ). The Master server aggregates users' gradients and produces a new model, denoted FURIA 1 , ready for use at the following day. This process is iteratively done once at the end of the day (e.g., midnight) for continuous learning of users' mobility behavior.

Compas'2020 : Parallélisme/ Architecture/ Système / Temps Réel MILC -Lyon, France, du 30 juin au 3 juillet 2020

Threat Model

In this work, the following assumptions are made. First, as EDEN uses FURIA's global model to assess its effectiveness, the latter requires a trusted proxy server, the so called Master Server. Its main task is to aggregate the local mobility updates of each participant and produce a joint updated model. The latter can be implemented in Trusted Execution Environment. Mobile user devices which participate collaboratively in building FURIA's global model are not trusted. This means that a malicious participant can use poisoning attacks to introduce backdoor functionality into the global model and infer sensitive information. Many current researchers are working on this aspect which is orthogonal to our research objectives [START_REF] Fung | Mitigating sybils in federated learning poisoning[END_REF]. In order to update FURIA's global model, users need to send their transformed mobility data at night to the Master Server. User devices are supposed to be charging, idle and connected to WIFI. Finally, the LBS is honest but curious. It responds to users' requests on one hand but on the other hand it may exploit the mobility data maliciously to infer personal information about the users.

Experimental Evaluation

Experimental Setup

EDEN is developed in Python by using Pytorch library [START_REF]PyTorch an open source machine learning library based on the torch library[END_REF] for developing the model and Pysyft [START_REF]PySyft is a python library for secure and private deep learning[END_REF] for emulating the federated protocols and participants. For data preprocessing, we use Scikit-learn [START_REF]Scikit-learn is a free software machine learning library for the python programming language[END_REF] to scale data, and S2Geometry [START_REF]S2 is a library for spherical geometry[END_REF] to form the features vector, by gathering mobility data on cells with edge length ranges from 850m to 1185m .

In our experiments, the training is done on multiple round, each round represents a day. We assume that users train the model each midnight. On this way, on day x we train the model on data collected the day before (i.e. day x-1) and we test the model on data collected that day. We use the simple gradient descent algorithm as an optimizer for the logistic regression, with a value of 0.001 for the learning rate parameter.

State-of-the-art LPPMs.In our scenario, we consider several existing LPPMs (location privacy protection mechanisms), and compare them to our federated learning-based protection mechanism. In order to apply LPPMs in our dataset, we used an open source library [START_REF] Primault | Accio : How to make location privacy experimentation open and easy[END_REF], with configurations described in Table 1. Mobility Dataset. In our experiments, we use the Privamo dataset which contains real mobility data of 48 users in Lyon [START_REF] Boutet | Uniqueness assessment of human mobility on multi-sensor datasets[END_REF]. We extracted the most active month (i.e. 30 days). For each user, we formed traces by gathering all mobility data in chunks of 30 minutes, which corresponds to a LBS that collects users' mobility data every 30 minutes.

LPPM LPPM's configuration

Geo-Indistinguishability (Geo-I) [START_REF] Andrés | Geoindistinguishability : Differential privacy for location-based systems[END_REF] 0.01, 0.005, 0.001 Promesse (PROM) [START_REF] Primault | Time distortion anonymization for the publication of mobility data with high utility[END_REF] 50 m, 100 m , 200 m Trilateration (TRL) [START_REF] Huang | Search locations safely and accurately : A location privacy protection algorithm with accurate service[END_REF] 1 km, 2 km, 3 km TABLE 1 -State-of-the-art LPPMs and their configurations

Comparison of EDEN with State-of-the-Art LPPMs

In the following, we run our algorithm attack on protected data and compute the attack success rate each day. Then, We compare EDEN with Geo-I, TRL, and PROM. Due to space constraints, we represent in Figure 2 only results regarding to the best configuration in term of privacy for each LPPM. We plot the curve associated to the attack over none obfuscated data (i.e. NOBF) as a reference for the protection efficiency. For each case, we chose one color : orange, blue, red,

Compas'2020 : Parallélisme/ Architecture/ Système / Temps Réel MILC -Lyon, France, du 30 juin au 3 juillet 2020 green and pink for NOBF, Geo-I, TRL, PROM and EDEN respectively. The attack success rate in our protection does not exceed 4%, as a comparison the accuracy for Geo-I range from 31% to 75%, for TRL is from 10% to 63% and for PROM it does not exceed 10%. EDEN protect 100% of the traces during 7 days and at 99% during 17 days. The accuracy decreases in all cases in some days because of the appearance of new users. Figure 3 presents the utility of data protected with state-of-the-Art LPPMs and compares it to EDEN. Here, utility metric is the area coverage of the protected data compared to the raw (i.e., non-protected) data. We consider three levels of utility : high utility if the area coverage is higher than 60%, utility if it is between 30% to 60%, and low in other cases. We observe that apart from the case of EDEN and PROM (configured with a 50 m distance), more than the half of protected data have a low utility. Furthermore, EDEN has the highest ratio of protected data with high utility.

Conclusion

In this paper we presented EDEN a solution that protects mobility data by choosing the best among a set of off-the-shelf LPPMs, without relying on a trusted proxy. Specifically, EDEN relies on federated learning to train a decentralized attack model called FURIA. This attack is then used locally along with a utility metric in order to choose the most appropriate LPPM for a given mobility trace. We evaluated EDEN by performing a set of experiments on real mobility traces over a period of one month. We used EDEN to choose the best LPPM and its corresponding configuration among nine possibilities. The results shown that EDEN outperforms individual LPPMs both in terms of privacy (better resilience against our FURIA decetralized attack) and in terms of data utility.

  FURIA : Federated User Re-Identification Attack X: set of features Y: set of labels (IDs) Ui: User "i"

FIGURE 1 -

 1

FIGURE 2 -

 2 FIGURE 2 -Attack success rate with EDEN vs. state-of-the-art LPPMs

FIGURE 3 -

 3 FIGURE 3 -Data utility with EDEN vs. state-of-the-art LPPMs


Zdeněk Dvořák
email: zdenek.dvorak@upol.cz

Harry Sokol

Sridhar Mani
email: sridhar.mani@einsteinmed.org

AhR, and Microbial Metabolite Interactions in the Intestine

Keywords:

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

Opinion

Drug Mimicry: Promiscuous Receptors PXR and AhR, and Microbial Metabolite Interactions in the Intestine Zdeněk Dvořák, 1, * Harry Sokol, 2,3,4 and Sridhar Mani 5,6,7, * Significant attrition limits drug discovery. The available chemical entities present with drug-like features contribute to this limitation. Using specific examples of promiscuous receptor-ligand interactions, a case is made for expanding the chemical space for drug-like molecules. These ligand-receptor interactions are poor candidates for the drug discovery process. However, provided herein are specific examples of ligand-receptor or transcription-factor interactions, namely, the pregnane X receptor (PXR) and the aryl hydrocarbon receptor (AhR), and its interactions with microbial metabolites. Discrete examples of microbial metabolite mimicry are shown to yield more potent and non-toxic therapeutic leads for pathophysiological conditions regulated by PXR and AhR. These examples underscore the opinion that microbial metabolite mimicry of promiscuous ligand-receptor interactions is warranted, and will likely expand the existing chemical space of drugs.

Expanding the Chemical Space of Drugs via Microbial Metabolite Mimicry of Promiscuous Metabolite-Receptor Interactions

Promiscuous receptor-ligand interactions, while on the one hand are problematic for necessary science investigations into receptor biology, maybe a treasure trove for investigating the bounds of chemical space available for that receptor [START_REF] Keah | A molecular recognition paradigm: promiscuity associated with the ligand-receptor interactions of the activin members of the TGF-beta superfamily[END_REF]. If the receptor per se, influences a biological phenomenon, then there is theoretical potential to develop more potent small molecules mimicking that chemical space [START_REF] Ekins | Challenges predicting ligand-receptor interactions of promiscuous proteins: the nuclear receptor PXR[END_REF][START_REF] Giani Tagliabue | Modeling the binding of diverse ligands within the Ah receptor ligand binding domain[END_REF]. To mine this concept, we start with orphan or adopted orphan nuclear receptors, as it is well established that these sets of receptors, and related ligand-activated transcription factors, have significant ligand promiscuity [START_REF] Sladek | What are nuclear receptor ligands?[END_REF].

Orphan Nuclear Receptors and Transcription Factors as Attractive Targets for Host Modulation of Physiology and Pathophysiology

The absence of a well-defined endogenous ligand defines 'orphan nuclear receptors'. Once the endogenous ligand is discovered, the terms, 'deorphanization' or 'adopted orphan receptors' are used. The most studied subgroups of the orphan nuclear receptors comprise xenobiotic receptors, and metabolic and energy sensors. Unlike nuclear hormone receptors, orphan nuclear receptors have a large ligand-binding domain, and they are promiscuous in binding a wide array of structurally unrelated compounds. Also, ligands of orphan nuclear receptors usually have a lower affinity and potency than nuclear hormone receptors. Therefore, orphan nuclear receptors (salient receptors are elaborated on later) are sometimes referred to as high-capacity and lowspecificity (affinity) receptors.

The discovery of the first xenosensor, the AhR, a ligand activated transcription factor, dates back to the early 1970s. The AhR is activated by several xenobiotics, including environmental

Highlights

The lack of chemical diversity in drug-like molecules is one important reason for drug attrition.

Microbial metabolite mimicry of promiscuous ligand-receptor interactions is warranted, and will likely expand the existing chemical space of drugs.

Prototypical host receptors that exhibit weak ligand interactions include xenobiotic nuclear receptors such as PXR and AhR.

Using PXR and AhR as exemplar xenobiotic receptors, discrete examples of microbial metabolite mimicry, are shown to yield more potent and nontoxic therapeutic leads, for pathophysiological conditions regulated by these receptors.

pollutants, plant polyphenolics, alkaloids, synthetic compounds, and drugs. The examples of endogenous AhR ligands are eicosanoids, indirubin, bilirubin, or 6-formylindolo [3,2-b]carbazole (FICZ) [START_REF] Denison | Activation of the aryl hydrocarbon receptor by structurally diverse exogenous and endogenous chemicals[END_REF]. Whereas the AhR was considered for a long time as a transcriptional mediator of xenoprotective and drug-metabolizing genes, there is a substantial body of evidence for the involvement of the AhR in many physiological processes, such as hematopoiesis, restorative neurogenesis, organ development, embryogenesis, and immunity [START_REF] Rothhammer | The aryl hydrocarbon receptor: an environmental sensor integrating immune responses in health and disease[END_REF]. Moreover, the AhR is a pivotal actor in the pathogenesis of cancer [START_REF] Murray | Aryl hydrocarbon receptor ligands in cancer: friend and foe[END_REF], intestinal inflammation [START_REF] Metidji | The environmental sensor AHR protects from inflammatory damage by maintaining intestinal stem cell homeostasis and barrier integrity[END_REF], hepatic steatosis [START_REF] Lee | A novel role for the dioxin receptor in fatty acid metabolism and hepatic steatosis[END_REF], and atopic dermatitis [START_REF] Peppers | A phase 2, randomized dose-finding study of tapinarof (GSK2894512 cream) for the treatment of atopic dermatitis[END_REF]. Given the broad and essential roles of the AhR in human physiology and pathophysiology, the attempts for therapeutic targeting of the AhR have emerged. Multiple reports proposed the AhR signaling pathway as a target for anticancer therapy, including a strategy of repurposing existing drugs [START_REF] Murray | Aryl hydrocarbon receptor ligands in cancer: friend and foe[END_REF][START_REF] Cheong | Targeting the IDO1/TDO2-KYN-AhR pathway for cancer immunotherapy -challenges and opportunities[END_REF]. Clinically approved AhR active drugs lansoprazole, omeprazole, raloxifene, flutamide, sulindac, tranilast, leflunomide, nimodipine, or mexeletine were suggested as off-target chemotherapeutics for the treatment of breast cancers [START_REF] Baker | The aryl hydrocarbon receptor (AhR) as a breast cancer drug target[END_REF]. Off targeting the AhR with tranilast was used in the treatment of atopic dermatitis [START_REF] Darakhshan | Tranilast: a review of its therapeutic applications[END_REF]. Topical application of coal tar induced skin barrier repair in atopic dermatitis via the AhR pathway [START_REF] Van Den Bogaard | Coal tar induces AHR-dependent skin barrier repair in atopic dermatitis[END_REF]. A topical cream containing tapinarof, a bacterial stilbenoid, and an AhR agonist, was examined, as an agent for atopic dermatitis [START_REF] Peppers | A phase 2, randomized dose-finding study of tapinarof (GSK2894512 cream) for the treatment of atopic dermatitis[END_REF] and plaque psoriasis [START_REF] Robbins | Phase 2, randomized dose-finding study of tapinarof (GSK2894512 cream) for the treatment of plaque psoriasis[END_REF]. The roles of the AhR in the pathogenesis of inflammatory bowel diseases are widely known [START_REF] Lamas | CARD9 impacts colitis by altering gut microbiota metabolism of tryptophan into aryl hydrocarbon receptor ligands[END_REF], and the insufficient or inappropriate activation of the AhR is associated with inflammatory conditions [START_REF] Metidji | The environmental sensor AHR protects from inflammatory damage by maintaining intestinal stem cell homeostasis and barrier integrity[END_REF]. Whereas the activation of the intestinal AhR by various xenobiotics was reported to trigger and to attenuate colitis, therapeutic targeting of the intestinal AhR was not applied to date. Interestingly, very few AhR antagonists described, and more recently, natural vitamins, folate, and B12, seem to function as endogenous AhR antagonists [START_REF] Kim | Vitamin B12 and folic acid alleviate symptoms of nutritional deficiency by antagonizing aryl hydrocarbon receptor[END_REF].

The PXR was initially described as an orphan nuclear receptor, activated by natural and synthetic steroids. In this context, it is important to recognize that the mouse and the human receptors respond to distinct ligands (e.g., the mouse receptor is activated by pregnenolone carbonitrile and the human receptor is activated by rifampicin), and therefore, displays species-specificity. In this context, with respect to intestinal microbial metabolites, indolyl-3-propionate (IPA) is a mouse PXR ligand, but not an efficient human PXR ligand. However, combinatorial indole metabolite mixtures can activate the human PXR receptor. These metabolites are produced in both rodent and human intestines. Thus, there is specificity in terms of the types of ligands found, as microbial metabolites and their ligand activation of PXR. For translation to human conditions, human PXR ligands also necessitate the use of humanized mice and these mice have been created by several groups and companies (e.g., Taconic). The PXR is activated by xenobiotics that induce drug-metabolizing enzymes, causing pharmacokinetic drug interactions, which is undesired [START_REF] Lehmann | The human orphan nuclear receptor PXR is activated by compounds that regulate CYP3A4 gene expression and cause drug interactions[END_REF]. Xenobiotic ligands of the PXR are structurally unrelated compounds, including drugs (e.g., rifampicin), natural compounds (e.g., hyperforin and solomonsterols) and environmental pollutants (e.g., phthalates and bisphenols) [START_REF] Chai | Drug discovery technologies to identify and characterize modulators of the pregnane X receptor and the constitutive androstane receptor[END_REF][START_REF] Chai | Strategies for developing pregnane X receptor antagonists: implications from metabolism to cancer[END_REF]. Currently, the PXR is 'deorphanized' and endogenous ligands such as bile acids, estrogens, or vitamin K2, were identified. Besides the central role in the regulation of xenobiotic-metabolizing enzymes, the PXR controls the intermediary metabolism of lipids, glucose, and bile acids, and it is involved in the etiology of various diseases such as diabetes, metabolic syndrome, cardiovascular pathologies, inflammatory bowel disease, acute kidney injury, neurological pathologies, and cancer [START_REF] Chai | Strategies for developing pregnane X receptor antagonists: implications from metabolism to cancer[END_REF]. Hence, the therapeutic targeting of the PXR with both agonists and antagonists is of particular interest [START_REF] Chai | Drug discovery technologies to identify and characterize modulators of the pregnane X receptor and the constitutive androstane receptor[END_REF][START_REF] Chai | Strategies for developing pregnane X receptor antagonists: implications from metabolism to cancer[END_REF].

The constitutive androstane receptor (CAR) is a 'sister' receptor of the PXR [START_REF] Forman | Androstane metabolites bind to and deactivate the nuclear receptor CAR-beta[END_REF], because they share and overlap ligands and activators, transcriptional partners, specific response elements in DNA, a set of target genes, and in general, biological functions [START_REF] Willson | PXR, CAR and drug metabolism[END_REF]. In contrast to other nuclear receptors, the CAR constitutively activates gene transcription. Initially, it was observed that the steroids androstanol and androstenol inhibit the constitutive activity of the CAR (inverse agonist effects) by a mechanism involving the promotion of coactivator release from the ligand-binding domain. Therefore, an unanticipated steroidal signaling pathway that functions in a manner opposite to that of the conventional nuclear receptor pathways was defined [START_REF] Forman | Androstane metabolites bind to and deactivate the nuclear receptor CAR-beta[END_REF]. Targeting the CAR in the therapy of metabolic diseases, cancer, diabetes, inflammatory, and liver diseases, were proposed [START_REF] Banerjee | Targeting xenobiotic receptors PXR and CAR in human diseases[END_REF].

The liver X receptor (LXR) exists in two forms, (i.e., LXRα and LXRβ), which have tissue-specific expression. The endogenous ligands of the LXRs are oxysterols, and intermediates of the mevalonate metabolic pathway. An example of the xenobiotic activator is synthetic compound GW3965. Physiological roles for the LXRs are the regulation of glucose, fatty acids, and cholesterol metabolism. Therefore, the LXRs are promising targets for the therapy of obesity, diabetes, and atherosclerosis [START_REF] Calkin | Transcriptional integration of metabolism by the nuclear sterol-activated receptors LXR and FXR[END_REF]. Therapeutic targeting of the LXRs, using dual or LXRα/LXRβ-selective ligands, is also considered in cancer [START_REF] Lin | Targeting liver X receptors in cancer therapeutics[END_REF], and inflammatory and neurodegenerative diseases [START_REF] Hong | Liver X receptors in lipid metabolism: opportunities for drug discovery[END_REF]. Oltipraz, an antagonist of the LXRα, reduced the liver fat content in patients with nonalcoholic fatty liver disease (NAFLD) in phase II clinical trial [START_REF] Kim | Randomised clinical trial: the efficacy and safety of oltipraz, a liver X receptor alpha-inhibitory dithiolethione in patients with non-alcoholic fatty liver disease[END_REF]. The LXRβ-selective agonist BMS-852927, increased reverse cholesterol transport pathways, but adverse effects, such as elevated LDL cholesterol and triglycerides, were reported [START_REF] Kirchgessner | Beneficial and adverse effects of an LXR agonist on human lipid and lipoprotein metabolism and circulating neutrophils[END_REF]. An improvement in barrier differentiation and lipids, was observed in patients with mild to moderate atopic dermatitis, topically administered with LXRβ-selective ligand VTP-38543 [START_REF] Czarnowicki | Effect of short-term liver X receptor activation on epidermal barrier features in mild to moderate atopic dermatitis: a randomized controlled trial[END_REF].

The farnesoid X receptor (FXR; NR1H4) was named after its first identified ligand farnesol. Endogenous ligands of the FXR are bile acids; hence the FXR is sometimes referred to as bile acid receptor. The physiological function of the FXR is the regulation of the metabolism of bile acids, fatty acids, triglycerides, and glucose. Consistently, FXR is an attractive target for the treatment of metabolic and hepatic diseases [START_REF] Calkin | Transcriptional integration of metabolism by the nuclear sterol-activated receptors LXR and FXR[END_REF][START_REF] Schaap | Bile acid receptors as targets for drug development[END_REF]. Several clinical trials were carried out with highly potent FXR synthetic agonist obeticholic acid, including those against type II diabetes and NAFLD [START_REF] Mudaliar | Efficacy and safety of the farnesoid X receptor agonist obeticholic acid in patients with type 2 diabetes and nonalcoholic fatty liver disease[END_REF], non-cirrhotic nonalcoholic steatohepatitis [START_REF] Neuschwander-Tetri | Farnesoid X nuclear receptor ligand obeticholic acid for non-cirrhotic, non-alcoholic steatohepatitis (FLINT): a multicentre, randomised, placebocontrolled trial[END_REF], primary sclerosing cholangitis [START_REF] Kowdley | A randomized, placebo-controlled, phase II study of obeticholic acid for primary sclerosing cholangitis[END_REF], and primary biliary cholangitis [START_REF] Nevens | Group PS. A placebo-controlled trial of obeticholic acid in primary biliary cholangitis[END_REF]. For the latter, FDA-approved drug Ocaliva® is available. The examples of other FXR agonists currently under clinical investigations are Cilofexor (Gilead Sciences), and PX-102 (Phenex Pharmaceuticals).

The peroxisome proliferator-activated receptors (PPARs) regulate glucose and lipid homeostasis, inflammation, proliferation, and differentiation [START_REF] Peters | The role of peroxisome proliferatoractivated receptors in carcinogenesis and chemoprevention[END_REF]. Fatty acids activate the PPARs and their derivatives, the examples of endogenous ligands for PPARα and PPARγ are oleoylethanolamide and 15-deoxy-prostaglandin J2, respectively. Xenobiotic ligands for the PPARs are various endocrine-disrupting chemicals that cause obesogenic effects. The therapeutic target for hypolipidemic drugs of fibrates class is PPARα, whereas antidiabetes type-II thiazolidinediones target PPARγ. Targeting PPARs in cancer therapy [START_REF] Peters | The role of peroxisome proliferatoractivated receptors in carcinogenesis and chemoprevention[END_REF] and diabetic microvascular damage [START_REF] Hiukka | PPARalpha: an emerging therapeutic target in diabetic microvascular damage[END_REF] was proposed.

Microbial Metabolites as Modulators of AhR, PXR, FXR, LXRs, and PPARs Receptors

Human gut microbiota produces a broad spectrum of metabolic products that interact with the host organism, and mediate beneficial health effects and pathogenicity. The microbial metabolites are structurally diverse compounds, including short and medium-chain fatty acids, polysaccharides, polyamines, formyl-peptides, bile acids, tryptophan metabolites, acylhomoserine lactones, gaseous substances (hydrogen sulfide, methane), toxins, and others [START_REF] Rooks | Gut microbiota, metabolites and host immunity[END_REF]. At cellular levels, the gut microbial metabolites interact mostly with G protein-coupled receptors and nuclear receptors, through which they mediate their biological effects both onsite and distantly.

The following examples illustrate how multiple microbial metabolites can serve as weak but promiscuous ligands, for a given nuclear receptor, and across ligand-activated transcription factor systems. In this context only PXR and AhR are highlighted, as the other receptors mentioned, LXR, FXR, and PPAR, are not, per se, promiscuous receptors. Metabolites of tryptophan, produced by human intestinal microbiota, were extensively studied as ligands of the AhR and mediators of intestinal health and disease. To date, numerous tryptophan catabolites were identified as low-affinity AhR ligands (agonist and antagonist), including indole, skatole, tryptamine, and a series of indolyl-3-(lactate, pyruvate, acrylate, propionate, acetate, aldehyde, acetamide, ethanol) ligands [START_REF] Roager | Microbial tryptophan catabolites in health and disease[END_REF][START_REF] Vyhlidalova | Gut microbial catabolites of tryptophan are ligands and agonists of the aryl hydrocarbon receptor: a detailed characterization[END_REF]. Ligand-dependent transcriptional activation of the AhR, was synergistically enhanced by short-chain fatty acids (SCFAs) [START_REF] Jin | Short chain fatty acids enhance aryl hydrocarbon (Ah) responsiveness in mouse colonocytes and caco-2 human colon cancer cells[END_REF]. Other examples of intestinal-microbiota derived AhR ligands comprise 2,8-dihydroxy-quinoline, urolithin A (formed from ellagitannins), and 1,4-dihydroxy-2-naphthoic acid [START_REF] Pernomian | The aryl hydrocarbon receptor (AHR) as a potential target for the control of intestinal inflammation: insights from an immune and bacteria sensor receptor[END_REF].

The most studied microbial PXR ligands were the indole products of tryptophan catabolism, including skatole [START_REF] Vyhlidalova | Differential activation of human pregnane X receptor PXR by isomeric mono-methylated indoles in intestinal and hepatic in vitro models[END_REF], indolyl-3-acetate (IAA), and IPA; the latter being reported to diminish intestinal inflammation through PXR pathway [START_REF] Venkatesh | Symbiotic bacterial metabolites regulate gastrointestinal barrier function via the xenobiotic sensor PXR and Toll-like receptor 4[END_REF]. Intestinal proinflammatory responses in the early stages of experimental necrotizing enterocolitis were attenuated by lithocholic acid, through targeting the PXR [START_REF] Huang | Targeting the PXR-TLR4 signaling pathway to reduce intestinal inflammation in an experimental model of necrotizing enterocolitis[END_REF]. Indirect effects of various xenobiotics such as statins or green tea polyphenols through PXR, were shown to result in dysregulation of intestinal SCFAs and bile acid composition. Whereas the involvement of CAR in the intestinal mucosal response to injury was demonstrated, direct proof of CAR targeting with microbial metabolites was not provided [START_REF] Hudson | Constitutive androstane receptor regulates the intestinal mucosal response to injury[END_REF].

The antioxidant tempol displayed antiobesity activity, by reducing intestinal genus Lactobacillus, leading to the accumulation of tauro-β-muricholic acid, which is the FXR antagonist [START_REF] Li | Microbiome remodelling leads to inhibition of intestinal farnesoid X receptor signalling and decreased obesity[END_REF]. Similarly, theabrownin-suppressed gut microbes, with bile-salt hydrolase activity, resulted in increased levels of taurochendoxycholic, tauroursodeoxycholic, glycochenodeoxycholic, and glycoursodeoxycholic acids, exhibiting hypolipidemic effects through inhibition of the intestinal FXR [START_REF] Huang | Theabrownin from Pu-erh tea attenuates hypercholesterolemia via modulation of gut microbiota and bile acid metabolism[END_REF]. Hypoglycemic activity of metformin was partly explained by shaping gut microbiota when Bacteroides fragilis diminished, and the levels of glycoursodeoxycholic acid, an FXR antagonist, increased [START_REF] Sun | Gut microbiota and intestinal FXR mediate the clinical benefits of metformin[END_REF]. Recently, a new class of bile acids with FXR agonist activity, produced by human gut microbiota, was identified. These acids are conjugated with atypical amino acids, yielding novel phenylalanocholic, tyrosocholic, and leucocholic acids [START_REF] Quinn | Global chemical effects of the microbiome include new bile-acid conjugations[END_REF]. Overall, gut bacterial bile acids are the agonist and antagonists of the FXR, that display their biological effects partly through this receptor; however, the activation of the PXR cannot be excluded. Interestingly, there are no available data about the interactions between LXR and gut microbial metabolites.

Several studies reported the activation of the intestinal PPARγ by SCFAs, mainly by propionate and butyrate [START_REF] Nepelska | Commensal gut bacteria modulate phosphorylation-dependent PPARgamma transcriptional activity in human intestinal epithelial cells[END_REF]. Depletion of butyrate-producing microbes by antibiotics reduced epithelial signaling through PPARγ [START_REF] Byndloss | Microbiota-activated PPAR-gamma signaling inhibits dysbiotic Enterobacteriaceae expansion[END_REF]. Probiotic bacteria producing conjugated linoleic acids locally in the gut, were shown to suppress colitis through targeting PPARγ [START_REF] Bassaganya-Riera | Probiotic bacteria produce conjugated linoleic acid locally in the gut that targets macrophage PPAR gamma to suppress colitis[END_REF].

Indole and Its Microbial Metabolites as Promiscuous Ligands, for a Given Receptor and across Receptor Systems

L-tryptophan catabolism, specifically via microbes in the intestine, is essential for host-microbe symbiosis [START_REF] Borghi | Tryptophan as a Central Hub for Host/ Microbial Symbiosis[END_REF][START_REF] Agus | Gut microbiota regulation of tryptophan metabolism in health and disease[END_REF] (Figure 1). Indole per se, has been extensively studied concerning AhR. Indole is a very weak PXR ligand [START_REF] Venkatesh | Symbiotic bacterial metabolites regulate gastrointestinal barrier function via the xenobiotic sensor PXR and Toll-like receptor 4[END_REF]. However, indoles (~1 mM) can also inhibit voltage-gated K (+) channels leading to acute stimulation of GLP-1 secretion from intestinal L cells; however, a more chronic effect of indoles reduces GLP-1 via inhibition of NADH dehydrogenases, and a slow loss of ATP [START_REF] Chimerel | Bacterial metabolite indole modulates incretin secretion from intestinal enteroendocrine L cells[END_REF]. Indole also has effects on bacteria, such that it causes spatial segregation of bacteria in the gut that resist invaders, but helps recruit beneficial organisms [START_REF] Yang | Biphasic chemotaxis of Escherichia coli to the microbiota metabolite indole[END_REF]. Additionally,

Trends in Pharmacological Sciences

Trends in Pharmacological Sciences, December 2020, Vol. 41, No. 12 903 indole can attenuate the IFN1 pathway, and limit graft-versus-host disease in mice that have undergone allogeneic bone marrow transplantation [START_REF] Swimm | Indoles derived from intestinal microbiota act via type I interferon signaling to limit graft-versus-host disease[END_REF]. IPA is a mouse PXR agonist which requires indole for the efficient activation of the human receptor [START_REF] Venkatesh | Symbiotic bacterial metabolites regulate gastrointestinal barrier function via the xenobiotic sensor PXR and Toll-like receptor 4[END_REF]. However, IPA is not an efficient AhR ligand [START_REF] Vyhlidalova | Gut microbial catabolites of tryptophan are ligands and agonists of the aryl hydrocarbon receptor: a detailed characterization[END_REF]. Indole metabolite, skatole (3-methylindole) is an efficient ligand for AhR, but also demonstrates AhR-independent activation of p38 in intestinal epithelial cells [START_REF] Kurata | Skatole regulates intestinal epithelial cellular functions through activating aryl hydrocarbon receptors and p38[END_REF]. Tryptamine mediates ionic flux in the intestines directly, via a 5-HT 4 receptor (5-HT 4 R), a G-protein-coupled receptor (GPCR) [START_REF] Bhattarai | Gut microbiota-produced tryptamine activates an epithelial G-protein-coupled receptor to increase colonic secretion[END_REF]. Indoleacrylic acid (IAC) and indole 3-pyruvic acid (IPY), suppress inflammation via actions on Nrf2 and AhR, respectively [START_REF] Wlodarska | Indoleacrylic acid produced by commensal peptostreptococcus species suppresses inflammation[END_REF][START_REF] Aoki | Indole-3-pyruvic acid, an aryl hydrocarbon receptor activator, suppresses experimental colitis in mice[END_REF]. These examples serve to illustrate the complexity of host receptor interactions within just one amino acid microbial catabolite family. Efficient metabolite-receptor interactions are usually in the micromolar range. The combinatorial effects of metabolites also have different outcomessome together act as agonists, while other agonists, in combination, could result in antagonism of the receptor(s) [START_REF] Jin | Microbiome-derived tryptophan metabolites and their aryl hydrocarbon receptor-dependent agonist and antagonist activities[END_REF]. Using the indole metabolite family as an example, further explored is how quantitative structure-activity relationships (QSAR), focused on PXR and AhR, may yield potential lead small molecules. It can be shown that these lead molecules are relatively safe, compared with known xenobiotic ligands for AhR and PXR, for in vivo applications. Since PXR and AhR have potent anti-inflammatory properties, there is a potential for developing the leads into therapeutic drugs for conditions such as inflammatory bowel disease.

Quantitative Structure-Activity Relationships Between Indole(s) and Receptors, PXR, and AhR

The correlation between a structure of xenobiotic and microbial indoles, and their capability to activate PXR and AhR receptors in vitro or in vivo is not straightforward, because there is mutual regulatory crosstalk between the AhR and the PXR. Also, numerous compounds are dual agonists for both receptors (Figure 1). Whereas indole is a very low potency (EC 50 ≈1000 μM) ligand of the PXR, its monomethylation at any position significantly increased its potency (EC 50 ≈25-75 μM) [START_REF] Vyhlidalova | Differential activation of human pregnane X receptor PXR by isomeric mono-methylated indoles in intestinal and hepatic in vitro models[END_REF]. The substitution of indole at position 3 with acetamide moiety led to medium-potency derivative, whereas the introduction of 2-hydroxyethyl, 2-aminoethyl, or acryloyl moiety, maintained feeble agonist effects. However, 3-substitutions with acetate, propionate, lactate, pyruvate, or aldehyde resulted rather in loss of even weak indole activity (manuscript submitted). The presence of indole-N-acetamide moiety was identified as a PXR-active pharmacophore of viral polymerase inhibitors [START_REF] Harper | Potent inhibitors of subgenomic hepatitis C virus RNA replication through optimization of indole-Nacetamide allosteric inhibitors of the viral NS5B polymerase[END_REF]. Interestingly, the combination of IPA and indole yielded much stronger intestinal anti-inflammatory effects via the PXR, as compared with the individual metabolites [START_REF] Venkatesh | Symbiotic bacterial metabolites regulate gastrointestinal barrier function via the xenobiotic sensor PXR and Toll-like receptor 4[END_REF]. We designed highly potent (EC 50 ≈1 μM) and efficacious PXR-selective agonists, by exploiting indole-based hybrid structure scaffolds.

The key PXR-active pharmacophore was 1-(phenylsulfonyl)-indol-2-yl backbone. The new introduction of the indol-2-yl group led to the acquisition of the AhR activity and the AhR/ PXR-dual agonists; the removal of the phenyl sulfonyl group resulted in AhR supra-agonist compounds [START_REF] Dvorak | Targeting the pregnane X receptor using microbial metabolite mimicry[END_REF]. Eleven metabolites (all being 3-substituted indoles, formed from dietary tryptophan by human intestinal microbiota) were identified as the ligands and activators of the AhR, substantially differing in their pharmacology parameters [START_REF] Vyhlidalova | Gut microbial catabolites of tryptophan are ligands and agonists of the aryl hydrocarbon receptor: a detailed characterization[END_REF]. Highly potent (EC 50 ≈1 nM) AhR-agonists, are based on 3-monosubstituted indoles [6-(indole-3-carbonyl)picolinonitrile and (indol-3-yl)(6-(trifluoromethyl)pyridin-2-yl)methanone] [START_REF] Chen | Modulation of lymphocyte-mediated tissue repair by rational design of heterocyclic aryl hydrocarbon receptor agonists[END_REF].

Examples of Mimicry of AhR and PXR Ligand Metabolites Derived from Microbes as Therapeutic Leads As discussed previously, it was demonstrated that the combination of indole, together with its metabolite IPA, activates the human PXR. PXR agonism regulates intestinal inflammation via the TLR4-NFκB pathway [START_REF] Venkatesh | Symbiotic bacterial metabolites regulate gastrointestinal barrier function via the xenobiotic sensor PXR and Toll-like receptor 4[END_REF]. Subsequently, based on the binding pharmacophore of indole and IPA on the PXR ligand-binding domain (LBD), a series of indole analogs arising for reaction intermediates and final products in PXR and AhR reporter assays, were tested [START_REF] Dvorak | Targeting the pregnane X receptor using microbial metabolite mimicry[END_REF]. In these studies, it was found that several indole analog synthetic intermediates (FKK1-10) were effective PXR agonists; however, only two of these, FFK2 and FKK9, had potent AhR agonist effects. In picking for selective PXR agonists, we focused on FKK5 and FKK6, based on the most favorable biophysical and biochemical activity profile. As a first example, FKK6 was further evaluated in human cell lines and intestinal organoid model systems of inflammation, and finally, in a mouse model of chemical colitis. The data suggest that FKK6 significantly abrogates indices of proinflammatory activity (IL8, lipocalin) in all model systems studied.

Furthermore, experiments in primary human intestinal biopsy tissues, obtained from patients with inflammatory bowel disease (IBD) ex vivo, are planned (personal communication, Dr Hao Li, Albert Einstein College of Medicine, Bronx, NY and Dr Efi Kokkotou, Beth Israel Deaconess Medical Center, Boston MA). A more focused approach is seen with AhR. Indole is an AhR agonist (EC 50 ~3 μM to 1.5 mM in luciferase reporter assays) [START_REF] Hubbard | Indole and tryptophan metabolism: endogenous and dietary routes to Ah receptor activation[END_REF][START_REF] Vyhlidalova | Gut microbial catabolites of tryptophan are ligands and agonists of the aryl hydrocarbon receptor: a detailed characterization[END_REF]. However, at Harvard, investigators screened a chemical library of indoles and indazoles, which resulted in developing a potent hit, PY109. This small molecule is a potent AhR agonist (EC 50 ~1.2 nM) and is a more potent and drug-like indole mimic that has demonstrated anti-inflammatory properties in mice [START_REF] Chen | Modulation of lymphocyte-mediated tissue repair by rational design of heterocyclic aryl hydrocarbon receptor agonists[END_REF]. PY109, when orally administered to mice, potently induced IL-22 and expanded tissue ILC3 and γδ T cell subpopulations. The investigators found several other hits like PY108, all capable of similar actions as PY109. These design and chemical synthesis principles may be extended to the microbial synthesis of potent alkaloids that serve as AhR ligands [START_REF] Mexia | A Biomimetic, one-step transformation of simple indolic compounds to malassezia-related alkaloids with high AhR potency and efficacy[END_REF][START_REF] Yu | A tryptophan metabolite of the skin microbiota attenuates inflammation in patients with atopic dermatitis through the aryl hydrocarbon receptor[END_REF].

Potential Application and Pitfalls of PXR and AhR Activating Microbial Metabolite Mimics in Human Diseases

It is tempting to dive into classical drug development, but it is prudent to understand the effects of chronically perturbing PXR/AhR signaling homeostasis. In the case of both receptors, PXR and AhR, hyperstimulation of either receptor has tissue-specific effects, some that will not be beneficial for drug discovery. For example, hyperstimulation of AhR systemically can lead to anemia, a fatty liver, and developmentally, a cleft palate [START_REF] Kim | Vitamin B12 and folic acid alleviate symptoms of nutritional deficiency by antagonizing aryl hydrocarbon receptor[END_REF][START_REF] Turski | AhR and IDO1 in pathogenesis of Covid-19 and the "Systemic AhR Activation Syndrome" a translational review and therapeutic perspectives[END_REF]. In this context, the aim of an AhR targeting treatment might be to restore the normal activation of the receptor, and not to overstimulate it. Similarly, systemic activation of PXR can have untoward effects with regards to unanticipated drug interactions, liver steatosis, and the potential to accelerate de novo malignancy [START_REF] Biswas | Elucidating the 'Jekyll and Hyde' nature of PXR: the case for discovering antagonists or allosteric antagonists[END_REF].

Furthermore, as with any receptor involved in homeostasis, tachyphylaxis of response is also possible. Hence, applications of principles of clinical pharmacology then become salient in the drug development process for agents targeting these types of receptors. For example, in the case of FKK6 and PXR, it is imperative to drive PXR activation, perhaps in an intermittent manner and with local confinement. So, if we target inflammatory bowel disease, the ideal setting is distal colitis, in which FKK6 formulations could only be delivered as enemas for local effect. Alternatively, intermittent rectal bolus could be applied to avoid consistent hyperstimulation of PXR. Thus, a combination of patient and disease selection, coupled with some regional approaches for drug delivery, might be the best option to move these potent agents forward.

Concluding Remarks

This opinion article covers the feasibility of developing potential therapeutic leads using the concept of chemical biomimicry. The biomimics are of 'weak', but host targeted microbial metabolites, where the exact relationship between the ligand and its cognate therapeutic target(s) are known. It is put forward that there are some prerequisites for this to happen. First, the chemistry of deriving mimics should be simple and well defined. There should be a resolution of enantiomers, as many of these receptors have enantiomer-specific interactions. Second, the relationship between the host target and the microbial metabolite should be well defined, preferably by direct binding studies. Third, there should be a firm ability to perform classical biochemistry studies with the target. Fourth, the analog will need to be tested in terms of its affinity for binding to the receptor. If possible, the crystal structures of the ligand-bound receptor should be ascertained. Fifth, the association, in terms of cause and effect between the host target and the physiology (or pathophysiology) under therapy, should be well characterized. However, even with these prerequisites, broader questions are looming (see Outstanding Questions). For example, it is unknown as to what the true extent of weak (and promiscuous) metabolite-receptor pairings is in humans, specifically those that modify host physiology or pathophysiology. If each is mined, how many can genuinely be subjected to the prerequisites mentioned in the previous text to further drug discovery? Drug discovery is already replete with ideas that expand chemical space simply based on the limits of chemistry per se [START_REF] Naveja | Finding constellations in chemical space through core analysis[END_REF][START_REF] Llanos | Exploration of the chemical space and its three historical regimes[END_REF]. Hence, an outstanding question is whether this could help the concept develop further, or if the concept of directed metabolite mimicry enlarges the chemical space. There is much to be learned from mining microbial metabolite-host receptor interactions, and there is a ray of hope that future work will exploit these interactions favorably.

Outstanding Questions

To what extent are host receptormetabolite pairings promiscuous and weak (affinity in the micro to millimolar range)?

How many of these interactions actually influence host physiology or pathophysiology ?

How many receptor-metabolite pairs that influence host physiology are actually amenable to chemical analog investigations? A major prerequisite for this would be that the ligand in question must be amenable to chemistry that is simple, and dramatically changes receptor interactions.

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906 Trends in Pharmacological Sciences, December 2020, Vol. 41, No. 12

Figure 1 .

 1 Figure 1. Tryptophan Intestinal Microbial Catabolites as Ligands and Agonists of Xenosensors, AhR and PXR. A chart summarizes molecular effects of known human intestinal microbial metabolites of tryptophan at AhR and PXR receptors. Heat-map shows a semi-quantitative profile of interactions: (i) Affinity, is a measure of compound binding at the receptor ligand binding domain (dissociation constant ≈K D ); (ii) Potency, refers to the concentration of the compound to produce 50% of maximal effect (half-maximal effective concentration ≈EC 50 ); (iii) Efficacy, is the maximum response that can be reached by the compound. Abbreviations: PXR, pregnane X receptor; AhR, aryl hydrocarbon receptor.

Trends in Pharmacological Sciences, December 2020, Vol. 41, No. 12

Trends in Pharmacological Sciences, December 2020, Vol. 41, No. 12

Trends in Pharmacological Sciences, December 2020, Vol. 41, No. 12 905

Trends in Pharmacological Sciences, December 2020, Vol. 41, No. 12

Acknowledgments Supported in part by The Peer Reviewed Medical Research Program -Investigator Initiated Research Award under Award No. W81XWH-17-1-0479; NIH grants (ES030197; CA 222469); Czech Health Research Council, grant number [NV19-05-00220] (to Z.D.).

Disclaimer Statement

S.M. and Z.D. have filed a patent application US 2019/0367475 A1 on PXR agonists and uses thereof for gut barrier dysfunction and treatment prevention.


Christophe Profeta

EXTREME VALUES OF CRITICAL AND SUBCRITICAL BRANCHING STABLE PROCESSES WITH POSITIVE JUMPS

Keywords: 2020 Mathematics Subject Classification. 60J80, 60G52, 60G51, 60G70 Branching stable process, Extreme values. 1

We consider a branching stable process with positive jumps, i.e. a continuous-time branching process in which the particles evolve independently as stable Lévy processes with positive jumps. Assuming the branching mechanism is critical or subcritical, we compute the asymptotics of the maximum location ever reached by a particle of the process.

1. Statement of the main result 1.1. Introduction. We consider a one-dimensional branching stable Lévy process. It is a continuoustime particle system in which individuals move according to independent α-stable Lévy processes, and split at exponential times into a random number of children.

More precisely, the process starts at time t = 0 with a single particle located at the origin.

(1) When not branching, each particle moves independently as a strictly α-stable Lévy process L with positive jumps. We refer to Bertoin [START_REF] Bertoin | Lévy Processes[END_REF]Chapter VIII] for an overview of such process.

In particular, the existence of positive jumps implies that the scaling parameter α and the skew parameter β satisfy the conditions :

α ∈ (0, 1) ∪ (1, 2) and β ∈ (-1, 1] or α = 1 and β = 0.

(2) Each particle lives for an exponentially distributed time of parameter 1, independently of the others. When it dies, it splits into a random number of children with distribution p = (p n ) n≥0 . We assume that the distribution p is non trivial (i.e. p 1 = 1) and admits moments of order at least 3, i.e. E[p 3 ] < +∞.

Such process may be constructed by first running a standard continuous-time Markov branching process Z (see for instance [2, Chapter III]), and then running independent α-stable Lévy processes (L (i) ) along the edges. With this notation, for each t > 0, the number of particles alive at time t is thus given by Z(t), and their locations by

L (1)
t , . . . , L (Z(t)) t

.

(1.1)

It is classic that when E[p] ≤ 1, the process will go extinct in finite time with probability one. As a consequence, one may define the overall maximum M α,β ever attained by one of the particle. The main result of the paper is the computation of the asymptotics of its tail distribution :

u(x) = P (M α,β ≥ x) .
Theorem 1. Let κ α,β > 0 be the constant such that :

P (L 1 ≥ x) ∼ x→+∞ κ α,β x -α . i) Assume that E[p] < 1.
The asymptotics of M α,β is given by

P(M α,β ≥ x) ∼ x→+∞ κ α,β 1 -E[p] x -α . ii) Assume that E[p] = 1.
The asymptotics of M α,β is given by

P(M α,β ≥ x) ∼ x→+∞ 2κ α,β σ 2 x -α/2 where σ 2 = Var(p) = +∞ n=2 n(n -1)p n .
Remark 2. The constant κ α,β may be computed explicitly, but depends on the normalization chosen for L. For instance, following Sato [8, p.88], if the characteristic exponent of L is given by :

ln E e iλLt = -c α,β |λ| α 1 -iβ tan πα 2 sgn(λ) for α = 1 -|λ| for α = 1 (1.2) with c α,β = cos πβ 2 min(α, 2 -α) , then, κ α,β =      1 π Γ(α) sin πα 2 (1 + β) if α < 1 1 π if α = 1 1 π Γ(α) sin π 2 (α + αβ -2β) if α > 1
The occurrence of σ 2 is a classic feature of such asymptotics, and was already observed by Fleischman & Sawyer [START_REF] Fleischman | Maximum geographic range of a mutant allele considered as a subtype of a Brownian branching random field[END_REF] in the case of Branching Brownian motion, or by Lalley & Shao [START_REF] Lalley | On the maximal displacement of critical branching random walk[END_REF] in the case of Branching Random Walks.

In the symmetric case (i.e. β = 0) and when p 0 = p 2 = 1 2 , Theorem 1 was first obtained by Lalley & Shao [START_REF] Lalley | Maximal displacement of critical branching symmetric stable processes[END_REF], who prove that

P (M α,0 ≥ x) ∼ x→+∞ 2 α x -α/2 (1.3)
when choosing the normalization

E e iλL 1 = exp - R (1 -e iλx ) dx |x| α+1 = exp - π Γ(α + 1) sin πα 2 |λ| α .
In this case, from Remark 2 and the recurrence formula for the Gamma function, the constant κ α,0 equals

κ α,0 = π Γ(α + 1) sin πα 2 × 1 π Γ(α) sin πα 2 = 1 α hence (1.
3) agrees with Theorem 1 since σ 2 = 1 for this binary branching mechanism.

The starting point of [START_REF] Lalley | Maximal displacement of critical branching symmetric stable processes[END_REF] was to show that u is the solution of a pseudo-differential equation involving the generator of the symmetric stable Lévy process. This in turn allows to obtain a Feynman-Kac representation of u, and the authors then deduce the asymptotics of Theorem 1 after a careful analysis of the jumps of the underlying stable Lévy process. We shall propose here another approach and rather work with an integral equation. L s the running supremum of the stable Lévy process L, and let e be a standard exponential random variable of parameter 1, independent from L.

Lemma 3. The function u is a solution of the integral equation :

u(x) = P (S e ≥ x) + E 1 {Se<x} u(x -L e ) -Φ 0 (x) + Φ R (x) (1.4)
where the main term Φ 0 is given by

Φ 0 (x) = (1 -E[p])E 1 {Se<x} u(x -L e ) + 1 2 E p 2 -p E 1 {Se<x} u 2 (x -L e )
and the remainder Φ R satisfies the bounds

0 ≤ Φ R (x) ≤ E[p 3 ]E 1 {Se<x} u 3 (x -L e ) . (1.5)
Proof. We start by applying the Markov property at the first branching event : α,β ) n∈N are independent copies of M α,β , which are also independent of the pair (L e , S e ). As a consequence, we obtain the integral equation :

P(M α,β < x) =
1 -u(x) = p 0 P (S e < x) + +∞ n=1 p n E 1 {Se<x} (1 -u(x -L e )) n .
(1.6)

Plugging the Taylor expansion with integral remainder

(1 -u) n = 1 -nu + n(n -1) 2 u 2 - n(n -1)(n -2) 6 u 3 1 0 (1 -ut) n-3 (1 -t) 2 dt in (1.6), we deduce that u(x) = P (S e ≥ x) + E 1 {Se<x} u(x -L e ) -Φ 0 (x) + Φ R (x)
where the remainder Φ R equals :

Φ R (x) = n≥3 p n n(n -1)(n -2) 6 1 0 E 1 {Se<x} u 3 (x -L e )(1 -u(x -L e )t) n-3 (1 -t) 2 dt.
Since 0 ≤ u(x) ≤ 1 for any x ≥ 0, the upper bound for Φ R is obtained by bounding the term to the power n -3 by 1.

Remark 4. Several terms in the equation (1.4) satisfied by u look like convolutions products. This will lead us to work with Laplace transforms and we thus set for a positive function f :

L[f ](λ) = +∞ 0 e -λx f (x)dx.
We shall repeatedly use in the following the standard Karamata's Tauberian theorem (see for instance Korevaar [START_REF] Korevaar | Tauberian theory. A century of developments[END_REF]Theorem 8.1]) which states that for γ ≥ 0 :

L[f ](λ) ∼ λ→0 1 λ γ h 1 λ ⇐⇒ x 0 f (z)dz ∼ x→+∞ 1 Γ(1 + γ) x γ h(x), (1.7)
where h is a slowly varying function.

Remark 5. It may be noted that Equation (1.4) involves the distributions of L e and S e . The key observation is the following equivalence of asymptotics for strictly α-stable Lévy processes with positive jumps, see Bertoin [3, p.221] :

P (L 1 ≥ x) ∼ x→+∞ P (S 1 ≥ x) ∼ x→+∞ κ α,β x -α . (1.8)
We shall indeed prove, through Laplace transforms, that as x → +∞ :

P (L 1 ≥ x) ≤ (1 -E[p])u(x) + 1 2 E p 2 -p u 2 (x) ≤ P (S 1 ≥ x)
and the result will follow from the equivalence of both asymptotics.

The remainder of the paper is devoted to the proof of Theorem 1 : Section 2 is dedicated to the case α ∈ (0, 1] and Section 3 to the case α ∈ (1, 2). The general idea of the proof is the same in both cases, and is composed of three steps. We shall first write down some general inequalities involving Laplace transforms, then apply these inequalities to Equation (1.4) and finally pass to the limit and apply Karamata's tauberian theorem. The main difference between both cases is the existence of the first moment of L e when α ∈ (1, 2). This will require us to make some extra computations, in order to remove the "first order" terms.

2. The case 0 < α ≤ 1 2.1. Preliminary lemma. We start by writing some general bounds for the Laplace transform of the terms appearing in Equation (1.4). In the forthcoming proofs, we will frequently use the case f = u and f = u 2 . To simplify the notation, we set :

η α (λ) = Γ(1 -α)λ α-1 if α < 1 -ln(λ) if α = 1
and L + e = max(0, L e ). Note that using (1.8), the Tauberian theorem and the scaling property to remove the independent exponential random variable e, we have the asymptotics

1 -E e -λSe λ ∼ λ↓0 1 -E e -λL + e λ ∼ λ↓0 κ α,β η α (λ).
(2.1) Lemma 6. Assume that α ∈ (0, 1] and let f : [0, +∞) → [0, +∞) be a positive and decreasing function. The following inequalities holds :

i) Upper bound : +∞ 0 e -λx E 1 {Se<x} f (x -L e ) dx ≤ E e -λSe L[f ](λ)
ii) Lower bound :

+∞ 0 e -λx E 1 {Se<x} f (x -L e ) dx ≥ E e -λL + e L[f ](λ) + f (0) E e -λSe -E e -λL + e λ -E 1 {Le<0} -Le 0 e -λz f (z)dz iii) Assume that lim x→+∞ f (x) = 0, then lim λ↓0 1 η α (λ) E 1 {Le<0} -Le 0 e -λz f (z)dz = 0.
Proof. The upper bound i) is a direct consequence of the a.s. inequality L e ≤ S e . Indeed, since f is decreasing, the Laplace transform of the convolution product yields

+∞ 0 e -λx E 1 {Se<x} f (x -L e ) dx ≤ +∞ 0 e -λx E 1 {Se<x} f (x -S e ) dx = E e -λSe L[f ](λ).
For the lower bound ii), still using that f is decreasing and L e ≤ S e a.s.,

+∞ 0 e -λx E 1 {Se<x} f (x -L e ) dx = +∞ 0 e -λx E (1 {Se<x} -1 {Le<x} )f (x -L e ) dx + +∞ 0 e -λx E 1 {Le<x} f (x -L e ) dx ≥ f (0)
+∞ 0 e -λx (P (S e < x) -P (L e < x)) dx

+ +∞ 0 e -λx E 1 {Le<x} f (x -L e ) dx = f (0) E e -λSe -E e -λL + e λ + +∞ 0 e -λx E 1 {Le<x} f (x -L e ) dx.
We next decompose the remaining integral according as whether L e < 0 or L e ≥ 0 :

+∞ 0 e -λx E 1 {Le<x} f (x -L e ) dx = E e -λLe 1 {Le≥0} L[f ](λ) + +∞ 0 e -λx E 1 {Le<0} f (x -L e ) dx.
Then, the Fubini-Tonelli theorem and a change of variable in the last integral yields :

+∞ 0 e -λx E 1 {Le<0} f (x -L e ) dx = E 1 {Le<0} +∞ -Le e -λz-λLe f (z)dz ≥ P (L e < 0) L[f ](λ) -E 1 {Le<0} -Le 0 e -λz f (z)dz
and the result follows by gathering the two previous terms :

+∞ 0 e -λx E 1 {Le<0} f (x -L e ) dx ≥ E e -λL + e L[f ](λ) -E 1 {Le<0} -Le 0 e -λz f (z)dz .
It remains to compute the limit iii). Notice first that if β = 1 (and thus α < 1), then the process L is a subordinator, hence L e ≥ 0 a.s and the expectation is null. We thus assume now that β ∈ (-1, 1). Let ε > 0 and take A ε large enough such that f (x) ≤ ε for x ≥ A ε . We decompose :

1 η α (λ) E 1 {Le<0} -Le 0 e -λz f (z)dz ≤ 1 η α (λ) E 1 {-Aε<Le<0} Aε 0 e -λz f (z)dz + ε η α (λ) E 1 {Le<-Aε} -Le Aε e -λz dz ≤ A ε η α (λ) f (0) + ε η α (λ) +∞ 0 e -λz P(-L e > z)dz --→ λ↓0 εκ α,-β
where the limit of the integral follows from (2.1) since -L is a stable Lévy process with parameter α and -β.

We now apply Lemma 6 to study Equation (1.4).

Analysis of Equation (1.4).

Lemma 7. The Laplace transform of Φ 0 -Φ R satisfies the following bounds for λ > 0 :

λ 1 -E[e -λSe ] L[Φ 0 -Φ R ](λ) ≤ 1 and λ 1 -E e -λL + e L[Φ 0 -Φ R ](λ) ≥ 1 -λL[u](λ) - λ 1 -E e -λL + e E 1 {Le<0}
-Le 0 e -λz u(z)dz .

Proof. Taking the Laplace transform of (1.4) and using Point i) of Lemma 6 with f = u, we deduce that

L[u](λ) ≤ 1 -E e -λSe λ + L[u](λ)E e -λSe -L[Φ 0 -Φ R ](λ)
which yields the upper bound

λ 1 -E [e -λSe ] L[Φ 0 -Φ R ](λ) ≤ 1 -λL[u](λ) ≤ 1.
To get the lower bound, we apply Point ii) of Lemma 6 still with f = u. Since u(0) = 1, this yields

L[u](λ) + L[Φ 0 -Φ R ](λ) ≥ 1 -E e -λL + e λ + E e -λL + e L[u](λ) -E 1 {Le<0} -Le 0 e -λz u(z)dz
which gives the lower bound, after rearranging the terms.

It remains now to study the limit of both expressions when λ ↓ 0. Then letting λ ↓ 0 in Lemma 7 and using (2.1) and Point iii) in Lemma 6 with f = u, we obtain

L[Φ 0 ](λ) -L[Φ R ](λ) ∼ λ↓0 κ α,β η α (λ). (2.2)
To simplify the notations, we set

ϕ 0 (x) = (1 -E[p])u(x) + E[p 2 -p] 2 u 2 (x) (2.3) so that L[Φ 0 ](λ) = +∞ 0 e -λx E 1 {Se<x} ϕ 0 (x -L e ) dx.
Note that the function ϕ 0 is positive and decreasing since E[p 2p] > 0, as p is an integer-valued random variable. On the one hand, observe that since Φ R ≥ 0, we deduce from Lemma 6 with

f = ϕ 0 that L[Φ 0 ](λ) -L[Φ R ](λ) ≤ L[ϕ 0 ](λ) hence, from (2.2), κ α,β ≤ lim inf λ↓0 1 η α (λ) L[ϕ 0 ](λ).
On the other hand, fix ε > 0 small enough. Since lim x→+∞ u(x) = 0, there exists A ε > 0 such that u(x) ≤ ε for x ≥ A ε . This implies that there exists K, independent of ε, such that :

u 3 (x) ≤ εKϕ 0 (x) for x ≥ A ε .
As a consequence, using (1.5) and Point i) of Lemma 6 with f = u 3 , we deduce

L[Φ R ](λ) ≤ E p 3 Aε 0 e -λx u 3 (x)dx + +∞ Aε e -λx u 3 (x)dx ≤ E p 3 (A ε + εKL[ϕ 0 ](λ)) .
Then, using Point ii) of Lemma 6 with f = ϕ 0 ,

L[Φ 0 ](λ) -L[Φ R ](λ) ≥ E e -λL + e L[ϕ 0 ](λ) + ϕ 0 (0) E e -λSe -E e -λL + e λ -E 1 {Le<0} -Le 0 e -λz ϕ 0 (z)dz -E p 3 (A ε + εKL[ϕ 0 ](λ)) . (2.4)
Dividing both sides by η α (λ) and applying Point iii) of Lemma 6, we deduce that

κ α,β ≥ 1 -εKE p 3 lim sup λ↓0 1 η α (λ) L[ϕ 0 ](λ).
Finally, we have thus proven that

L[ϕ 0 ](λ) ∼ λ↓0 κ α,β η α (λ)
hence, by the Tauberian theorem,

x 0 ϕ 0 (z)dz ∼ x→+∞ κ α,β Γ(2 -α) η α 1 x .
The result now follows by differentiation, since ϕ 0 is decreasing.

3. The case 1 < α < 2

In this case, the existence of the first moment of S e prevents us from using directly the Tauberian theorem as in the previous case. Indeed, when α ∈ (1, 2), letting λ ↓ 0 in the first inequality of Lemma 7, one obtains :

lim sup λ↓0 L[Φ 0 ](λ) -L[Φ R ](λ) ≤ E[S e ].
Going back to (2.4), this implies that

1 -εKE p 3 lim sup λ↓0 L[ϕ 0 ](λ) ≤ (1 + ϕ 0 (0))E[S e ] + E[p 3 ]A ε -E[1 {Le<0} L e ]ϕ 0 (0) < +∞ (3.1)
i.e. we can only deduce, by the monotone convergence theorem, that

i) if E[p] < 1, +∞ 0 u(x)dx = E[M α,β ] < +∞, (3.2) ii) while if E[p] = 1, +∞ 0 u 2 (x)dx < +∞.
We shall thus made a technical modification of the previous proof, and write down some new inequalities. Proof. The proof of Lemma 8 is similar to that of Lemma 6. Point i) follows from the decomposition x = x -S e + S e , using the fact that f is decreasing and computing the Laplace transforms of the convolution products. Point ii) follows similarly from the decomposition x = x -L e + L e , by separating the case L e < 0 and L e ≥ 0. Finally, for Point iii), observe first that if β = 1, then the random variable (-L e ) + admits exponential moments, hence

λ 2-α E 1 {Le<0} -Le 0 e -λz zf (z)dz ≤ λ 2-α f (0)E 1 {Le<0} L 2 e --→ λ↓0 0.
Take now β ∈ (-1, 1) and let ε > 0. By assumption, there exists A ε > 0 such that f (x) ≤ ε for x ≥ A ε . We then decompose :

λ 2-α E 1 {Le<0} -Le 0 e -λz zf (z)dz ≤ λ 2-α E 1 {-Aε<Le<0} Aε 0 e -λz zf (z)dz + ελ 2-α E 1 {Le<-Aε} -Le Aε e -λz zdz ≤ λ 2-α A 2 ε f (0) + ελ 2-α +∞ 0 e -λz zP(-L e > z)dz --→ λ↓0 εκ α,-β
where the limit of the integral follows as before from (2.1) and the Tauberian theorem.

We now apply Lemma 8 to the equation (1.4) satisfied by u.

3.2. Analysis of Equation (1.4) in the case α ∈ (1, 2).

Lemma 9. The Laplace transform of x(Φ 0 -Φ R ) satisfies the following bounds :

λ 2 1 -E [e -λSe ] -λE [S e e -λSe ] L[x(Φ 0 -Φ R )](λ) ≤ 1 + λ 2 E S e e -λSe 1 -E [e -λSe ] -λE [S e e -λSe ] L[u](λ)
and

λ 2 1 -E e -λL + e -λE L + e e -λL + e L[x(Φ 0 -Φ R )](λ) ≥ 1 -λ 2 L[xu](λ) - λ 2 Ξ(λ) 1 -E e -λL + e -λE L + e e -λL + e where Ξ(λ) = E 1 {Le<0} -Le 0 e -λz zu(z)dz + E 1 {Le<0} (-L e ) L[u](λ).
Proof. We first multiply (1.4) by x before taking the Laplace transform of both sides:

L[xu](λ) = +∞ 0 e -λx xP (S e ≥ x) dx + +∞ 0 e -λx xE 1 {Se<x} u(x -L e ) dx -L[x(Φ 0 -Φ R )](λ).
Integrating by parts the first term on the right-hand side, we have

+∞ 0 e -λx xP (S e ≥ x) dx = 1 -E e -λSe -λE S e e -λSe λ 2 ∼ λ↓0 κ α,β Γ(2 -α)λ α-2 (3.3)
from (2.1) and the Tauberian theorem. To get the upper bound, we apply Point i) of Lemma 8 with f = u :

L[x(Φ 0 -Φ R )](λ) + L[xu](λ) ≤ 1 -E e -λSe -λE S e e -λSe λ 2 + E e -λSe L[xu](λ) + E S e e -λSe L[u](λ).
Adding λE[S e e -λSe ]L[xu](λ) on the right-hand side and rearranging the terms yields the announced upper bound.

Similarly, to get the lower bound, we apply Point ii) of Lemma 8 with f = u : where the remainder R(λ) is given by :

L[x(Φ 0 -Φ R )](λ) + L[xu](λ) ≥ 1 -E e -λL + e -λE L + e e -λL + e λ 2 + E e -λL + e L[xu](λ) + E L e e -λL + e L[u](λ) -E 1 {Le<0} -Le
R(λ) = E 1 {Le<0} -Le 0 e -λz zu(z)dz -E 1 {Le<0} L e L[u](λ) + E L + e e -λL + e (λL[xu](λ) -L[u](λ)) .
Note that the last term on the right-hand side is negative and may thus be removed since, integrating by parts,

L[u](λ) -λL[xu](λ) = E M α,β e -λM α,β ≥ 0,
hence R(λ) ≤ Ξ(λ) which proves the lower bound.

3.3. Proof of Theorem 1 when α > 1. We now want to let λ ↓ 0 in Lemma 9. Notice first that, thanks to (2.1), the first terms on the left-hand side of both inequalities will converge towards the same quantity, which is given by (3.3). Also, by the monotone convergence theorem As a consequence, it only remains to show that

lim λ↓0 λ 2-α L[u](λ) = 0. (3.4) When E[p] < 1, this is a direct consequence of (3.2) since L[u](λ) ≤ E[M α,β ] < +∞.
The situation is trickier when E[p] = 1, and we shall rely on the following Lemma, which gives an a priori bound on u.

Lemma 10. Assume that α > 1 and E[p] = 1. Then, there exists a constant C > 0 such that for x large enough,

P (M α,β ≥ x) ≤ Cx -α/2 .
Proof. We mimic the arguments of [START_REF] Lalley | Maximal displacement of critical branching symmetric stable processes[END_REF]. Recall that since α > 1, the positivity parameter

ρ = P(L 1 ≥ 0) of L belongs to the interval [1 -1 α , 1 α ]. Denote by M (t)
α,β the maximum of the branching stable process on the interval [0, t] and by M (t) α,β the maximum of the branching stable process on the interval [t, +∞]. From the construction given in (1.1), we first deduce that

P (M α,β ≥ x) ≤ P M (t) α,β ≥ x + P M (t) α,β ≥ x ≤ P M (t) α,β ≥ x + P (Z(t) ≥ 1) .
Let us now denote by T x the first time at which a particle of the branching process reaches the level x. Then, by applying the Markov property at T x , we deduce that conditionally to {T x ≤ t} = {M (t) α,β ≥ x}, the expected number of particles above x at time t is greater than ρ :

E   Z(t) i=1 1 {L (i) t ≥x} M (t) α,β ≥ x   ≥ ρ ≥ 1 - 1 α which implies that P M (t) α,β ≥ x ≤ α α -1 E   Z(t) i=1 1 {L (i) t ≥x}   ≤ α α -1 E[Z(t)]P (L t ≥ x) = α α -1 P t 1/α L 1 ≥ x
since Z is independent from the positions (L (i) ), and E[Z(t)] = 1 for all t ≥ 0. Taking t = x α/2 , we have thus proven that

P (M α,β ≥ x) ≤ α α -1 P L 1 ≥ √ x + P Z(x α/2 ) ≥ 1
and the result follows by using (2.1) and the Kolmogorov's theorem, which states that

P (Z(t) ≥ 1) ∼ t→+∞ 2 σ 2 t see for instance Asmussen & Hering [1, Theorem 2.6].
We now come back to the limit (3.4). Applying Lemma 10 and the Tauberian theorem, we deduce that for λ small enough, there exists a constant C > 0 such that

λ 2-α L[u](λ) ≤ Cλ 1-α 2 --→ λ↓0 0 since α ∈ (1, 2
). As a consequence, by letting λ ↓ 0 in Lemma 9, we obtain the asymptotics

L[xΦ 0 ](λ) -L[xΦ R ](λ) ∼ λ↓0 κ α,β Γ(2 -α)λ α-2 .
The remainder of the proof is now similar to the case α ∈ (0, 1]. First, applying Point i) of Lemma 8 with f = ϕ 0 and using that Φ R ≥ 0, we deduce that and the result now follows by differentiation, since ϕ 0 is decreasing, see Lemma 11 in the Appendix.

L[xΦ 0 ](λ) -L[xΦ R ](λ) ≤ L[xϕ 0 ](λ) + E[S e ]L[ϕ 0 ](λ) which implies the lower bound lim inf λ↓0 λ 2-α L[xϕ 0 ](λ) ≥ κ α,β Γ(2 -α).

Appendix

We briefly prove the following lemma, which allows to differentiate an asymptotics :

Lemma 11. Assume that u is a positive and decreasing function and γ ∈ (0, 1). Then, for any ξ ≥ 0 :

x 0 z ξ u(z)dz ∼ x→+∞ x γ ⇐⇒ u(x) ∼ x→+∞ γx γ-1-ξ
Proof. We only prove the implication =⇒ as its converse is classic. Since u is decreasing, we have for h > 0 : Fix ε > 0. Assuming that x is large enough, we deduce from the assumption that (x + h) γx γ -2ε((x + h) γ + x γ ) ≤ h(x + h) ξ u(x).

Dividing both sides by h, we further obtain since γ ∈ (0, 1) :

γ(x + h) γ-1 - 2ε((x + h) γ + x γ ) h ≤ (x + h) ξ u(x).
We now set h = √ εx, divide both sides by γx γ-1 and let x → +∞. This yields :

(1 + √ ε) γ-1 -2 √ ε (1 + √ ε) γ + 1 γ ≤ 1 + √ ε ξ lim inf x→+∞ x ξ u(x) γx γ-1
i.e., letting ε ↓ 0,

1 ≤ lim inf x→+∞ x ξ u(x) γx γ-1 .
The other bound is obtained similarly, by starting from

1. 2 .

 2 An integral equation for u. We start by writing down the integral equation satisfied by the function u. Let us denote by S t = sup s∈[0,t]

  p 0 P (S e < x) + +∞ n=1 p n P S e < x, L e + M (1) α,β < x, . . . , L e + M

2. 3 .

 3 Proof of Theorem 1 when α ≤ 1. Notice first that from a change of variable and the monotone convergence theorem,

3. 1 .Lemma 8 . 0 eλ 2 -α E 1 {Le<0} -Le 0 e

 18020 Preliminary lemma. Assume that α ∈ (1, 2) and let f : [0, +∞) → [0, +∞) be a positive and decreasing function. We write xf for the function x → xf (x). Then, the following inequalities hold : i) Upper bound :+∞ 0 e -λx xE 1 {Se<x} f (x -L e ) dx ≤ E e -λSe L[xf ](λ) + E S e e -λSe L[f ](λ).ii) Lower bound :+∞ 0 e -λx xE 1 {Se<x} f (x -L e ) dx ≥ f (0) +∞ 0 e -λx x (P (L e ≥ x) -P (S e ≥ x)) dx + +∞ 0 e -λx xE 1 {Le<x} f (x -L e ) dxand +∞ -λx xE 1 {Le<x} f (x -L e ) dx ≥ E e -λL + e L[xf ](λ) + E L e e -λL + e L[f ](λ) -E 1 {Le<0} -Le 0 e -λz zf (z)dz . iii) Assume that lim x→+∞ f (x) = 0, then lim λ↓0 -λz zf (z)dz = 0.

0e 2 1- 2 1-

 22 -λz zu(z)dz Adding λE L + e e -λL + e L[xu](λ) in both sides and rearranging the terms, we obtain λ E e -λL + e -λE L + e e -λL + e L[x(Φ 0 -Φ R )](λ) ≥ 1λ 2 L[xu](λ) -λ E e -λL + e -λE L + e e -λL + e R(λ)

λ 2 Le

 2 Point iii) of Lemma 8 with f = u, -λz zu(z)dz = 0.

Next, we deduce

  from Point ii) of Lemma 8 with f = ϕ 0 that L[xΦ 0 ](λ) ≥ ϕ 0 (0) +∞ 0 e -λx x (P (L e ≥ x) -P (S e ≥ x)) dx + +∞ 0 e -λx xE 1 {Le<x} ϕ 0 (x -L e ) dx where +∞ 0 e -λx xE 1 {Le<x} ϕ 0 (x -L e ) dx ≥ E e -λL + e L[xϕ 0 ](λ) -E 1 {Le<0} -Le 0 e -λz zϕ 0 (z)dz + E L e e -λL + e L[ϕ 0 ](λ).Take ε > 0. As before, since lim x→+∞ u(x) = 0, there exists A ε > 0 and K > 0 such that u(x) ≤ ε andu 3 (x) ≤ εKϕ 0 (x) for x ≥ A ε .As a consequence, using (1.5) and Point i) of Lemma 8 with f = u 3 , we deduceL[xΦ R ](λ) ≤ E p 3 L[xu 3 ](λ) + E [S e ] L[u 3 ](λ) ≤ E p 3 A 2 ε + E [S e ] A ε + εK (L[xϕ 0 ](λ) + E [S e ] L[ϕ 0 ](λ)) ≤ C(1 + εL[xϕ 0 ](λ))for some constant C large enough since L[ϕ 0 ](λ) ≤ L[ϕ 0 ](0) < +∞ from (3.1). Then, using Point iii) of Lemma 8 with f = ϕ 0 , we obtain(1 -εC) lim sup λ↓0 L[xϕ 0 ](λ) ≤ κ α,β Γ(2α) which implies that L[xϕ 0 ](λ) ∼ λ↓0 κ α,β Γ(2α)λ α-2 .Finally, by the Tauberian theorem,

z

  ξ u(z)dz ≤ h(x + h) ξ u(x).

z

  ξ u(z)dz.