Hagai Abeliovich

Jayanta Debnath

Wen-Xing Ding

William T Jackson

Do-Hyung Kim

Daniel J Klionsky

Nicholas Ktistakis

Marta Margeta

Christian Münz

Morten Petersen

Junichi Sadoshima

Isabelle Vergne

Affiliations

Where is the field of autophagy research heading?

In this editors' corner, the section editors were asked to indicate where they see the autophagy field heading and to suggest what they consider to be key unanswered questions in their specialty area.

Hagai Abeliovich

Mitophagy

The selective degradation of mitochondria through macroautophagy, usually abbreviated as mitophagy, is a central housekeeping function in most eukaryotic cells. Defects in mitophagy have been linked to degenerative and metabolic disease states, and induction of mitophagy plays a role in developmental transitions in metazoans. Broadly, two types of molecular mechanisms have been linked to mitophagy. In one type, ubiquitination of outer and inner mitochondrial membrane proteins is linked to the autophagic machinery via bifunctional receptor proteins that interact both with ubiquitin (or phospho-ubiquitin) as well as with elements of the autophagic machinery such as LC3/Atg8. In the second type, a more classical "autophagy receptor" is localized to the target mitochondrion and directly binds elements of the autophagic machinery. While the mechanism of the first class, typified by the PINK1-PRKN axis, has been widely studied at the molecular level, much less is known about the mechanisms and function of the second, ubiquitin-independent, class of mitophagy pathways.

I think that these are exciting times in mitophagy research because there are so many unanswered questions. The following represent a sample: 1. Why is there such a proliferation of mitophagy receptors? Do these reflect cell-type or tissue specificity, specificity to mitochondrial subtypes, or specificity to the activation trigger? 2. Does the role of mitophagy in developmental transitions simply reflect a metabolic shift between respiratory and glycolytic lifestyles, or are there more subtle and complicated explanations?

3. Is quality control mitophagy a random event that is triggered by stochastic malfunction of a single mitochondrion? If so, then is the cell sacrificing a plethora of perfectly functioning molecules just to get rid of a few bad apples (the "Sodom and Gomorrah" paradox)?

4. Is mitophagy linked to mitochondrial heterogeneity? Mitochondrial heterogeneity is an acknowledged yet understudied phenomenon wherein the same cell may harbor mitochondria with different compositions, physiological functions, etc. Do different types of mitophagy act on different subpo pulations within the mitochondrial network?

What is the relationship between mitochondrial fission fusion dynamics and mitophagy?

The number of opinions on this question seems to be at least equal to the number of papers on the subject, with opinions ranging from discounting fission-fusion dynamics as a driving or permissive factor altogether, to arguments that fission is essential for mitophagy or that it is even a rate limiting step. [START_REF] Kitada | Autophagy in metabolic disease and ageing[END_REF]. What is the interplay between mitophagy and other mitochondrial quality control pathways? 7. Can pharmacological modulation of mitophagy play a therapeutic role in treating mitochondrial pathologies?

Jayanta Debnath

Cancer

Despite continued clinical interest in targeting macroautophagy/autophagy to treat cancer, many uncertainties remain due to the multifaceted roles that autophagy plays during initiation, progression and metastasis [START_REF] Amaravadi | Targeting autophagy in cancer: recent advances and future directions[END_REF][START_REF] Liu | The evolving, multifaceted roles of autophagy in cancer[END_REF]. Most attention in recent years has focused on inhibiting either autophagy or lysosomal function in advanced tumors due to the critical roles that autophagy plays in tumor cell survival and fitness, including nutrient scavenging, cell growth and metabolic adaptation. Indeed, certain tumors, most notably those with oncogenic mutations that rewire cellular metabolism (e.g., mutant RAS, BRAF, STK11/LKB) have proven highly sensitive to autophagy inhibition in diverse in vivo preclinical models [START_REF] Liu | The evolving, multifaceted roles of autophagy in cancer[END_REF]. Nevertheless, we have also learned that tumor cells rapidly develop resistance to autophagy inhibition [START_REF] Towers | Cancer cells upregulate NRF2 signaling to adapt to autophagy inhibition[END_REF] and that in certain advanced cancers, autophagy inhibition can unexpectedly promote metastatic outgrowth and recurrence [START_REF] Marsh | The pleiotropic functions of autophagy in metastasis[END_REF]. Hence, over the upcoming years, two important areas for future research will be to further scrutinize how tumor cells adapt to autophagy inhibition and to delineate strategies to target autophagy without untoward effects on metastasis.

Thinking beyond autophagy in tumor cells themselves, one of the most exciting developments in recent years has been the increasing appreciation of the roles that autophagy in host tissues plays in modulating cancer progression, response to therapy, and the immune recognition of tumors [START_REF] Amaravadi | Targeting autophagy in cancer: recent advances and future directions[END_REF][START_REF] Xia | Autophagy in tumour immunity and therapy[END_REF].

A number of studies have uncovered new functions for autophagy in the control of host-tumor metabolic exchange, the regulation of tumor cell immunity, and in the generation of stromal microenvironments permissive for tumor growth. Further defining these cardinal features of the host autophagic response to tumors and determining how they can be leveraged for therapeutic benefit remains an important area for investigation in the upcoming years.

Finally, from a therapeutic standpoint, the repurposing of hydroxychloroquine in clinical trials over the last decade has broached the promise of autophagy inhibition as a therapeutic strategy but also revealed its pharmacological limitations in humans [START_REF] Amaravadi | Targeting autophagy in cancer: recent advances and future directions[END_REF]. Overall, these studies point to two fundamental needs in the upcoming years in order to effectively target autophagy in cancer patients.

First, we require new chemical matter to specifically stimulate or inhibit autophagy in humans and to further understand how to best utilize such agents therapeutically during the various stages of cancer progression. Second, we require noninvasive strategies to monitor autophagy in cancer patients.

Wen-Xing Ding

Metabolism

As a catabolic pathway for proteins, lipids, carbohydrates and nucleic acids to produce amino acids, fatty acids, sugars and nucleosides, macroautophagy/autophagy is a major contributor to cellular metabolism for cell survival and remodeling. As such, autophagy has been implicated in normal development, physiology and various metabolic diseases such as insulin resistance, obesity, diabetes, nonalcoholic fatty liver disease (NAFLD), neurodegenerative disease and cancer [START_REF] Kitada | Autophagy in metabolic disease and ageing[END_REF][START_REF] Qian | Autophagy in liver diseases: a review[END_REF][START_REF] Rabinowitz | Autophagy and metabolism[END_REF]. While the evidence of autophagy in regulating metabolism is compelling, there are several unanswered key questions for autophagy in metabolism and metabolic diseases for future studies.

Autophagy is a highly dynamic process. The breakdown of metabolites such as amino acids and fatty acids from autophagic degradation also generates potent autophagy inhibitors, which thus establish a feedback inhibition loop on autophagy. Whether and how the metabolite-mediated feedback inhibition on autophagy fine tunes the balance of autophagy for the homeostasis of metabolic tissues has not been well elucidated.

Despite decades of research, safe pharmacological modulators (inducers and inhibitors) of autophagy in the prevention and treatment of metabolic diseases are still lacking. Some autophagy inducers such as metformin, rapamycin, trehalose and imatinib or autophagy inhibitors such as chloroquine and hydroxychloroquine, have been tested on animal models of diabetes, NAFLD and cancer [START_REF] Galluzzi | Pharmacological modulation of autophagy: therapeutic potential and persisting obstacles[END_REF]. However, these autophagy modulators may have multiple other targets in addition to targeting autophagy. High-throughput screening with followup validation of new autophagy modulators using genetic autophagy mouse models that have high (rubcn knockout mice) or low (Becn1+/-or tissue-specific atg5 knockout mice) autophagy activity in metabolic tissues may be helpful to identify more specific safe autophagy modulators for preventing and treating metabolic diseases.

Whereas activation of autophagy is generally beneficial for most metabolic tissues, increased autophagy may favor cancer cell survival and proliferation [START_REF] White | The role for autophagy in cancer[END_REF]. Defective autophagy in hepatocytes causes accumulation of damaged mitochondria and dysregulation of lipid and xenobiotic metabolism resulting in hepatomegaly, liver injury and spontaneous liver cancer [START_REF] Li | Impaired fasting-induced adaptive lipid droplet biogenesis in liverspecific atg5-deficient mouse liver is mediated by persistent nuclear factor-like 2 activation[END_REF][START_REF] Ni | Nrf2 promotes the development of fibrosis and tumorigenesis in mice with defective hepatic autophagy[END_REF][START_REF] Singh | Autophagy regulates lipid metabolism[END_REF][START_REF] Takamura | Autophagy-deficient mice develop multiple liver tumors[END_REF]. However, autophagy favors the activation of hepatic stellate cells that contributes to liver fibrosis [START_REF] Hernandez-Gea | Autophagy releases lipid that promotes fibrogenesis by activated hepatic stellate cells in mice and in human tissues[END_REF][START_REF] Thoen | A role for autophagy during hepatic stellate cell activation[END_REF], a process that also results in the accumulation of excess extracellular matrix proteins and collagen in many chronic liver diseases. How to identify tissue or cell-specific autophagy modulators is a challenging task for future studies.

There are dynamic cross-talks and communications between different metabolic tissues and/or organs via circulating endocrine hormones, cytokines, metabolites and exosomes secreted from these tissues. Disrupted or altered organ-organ communications have bene implicated in metabolic diseases [START_REF] Castillo-Armengol | Inter-organ communication: a gatekeeper for metabolic health[END_REF]. Secretory autophagy plays a critical role in facilitating unconventional secretion of cytosolic cargos for intracellular and organ-organ crosstalk [START_REF] Ponpuak | Secretory autophagy[END_REF]. How cells differentially regulate degradative autophagy versus secretory autophagy is another demanding question to be investigated.

William T. Jackson

Virology

For years the virology field has struggled with understanding how viruses interact with the machinery of macroautophagy/ autophagy. Frequently, the literature will claim a particular virus induces "autophagy," or benefits from "autophagy." If we use the accepted definition of autophagy to specifically mean "degradation of cytoplasmic contents," then very few, possibly no, viruses allow true autophagy to occur. Some viruses have evolved to inhibit autophagy because it is a threat, a way to clear cytosolic viruses and virus components from the cytosol. Most of these viruses, especially DNA viruses, actively repress autophagy initiation as infection progresses. Others prevent degradation by benefiting from specific aspects of the pathway. Many RNA viruses use the autophagic machinery as a source of lipids and membranes providing scaffolds for RNA replication, vesicular refuges for capsid assembly and maturation cleavages, and trafficking vehicles for non-lytic exit of virus. This release mechanism involves release of the inner membrane vesicle, filled with virus, through fusion of the outer autophagosomal membrane with the plasma membrane. However, these viruses often inhibit acidification of autophagosomes, lysosome fusion, and even cargo loading. In some cases, the relationship of autophagy to a given virus changes depending on whether autophagy is induced before (anti-viral) or after (pro-viral) infection. Presumably this is because the pool of autophagic resources is limited, and pre-inducing autophagy depletes them and makes them unavailable to the virus, curbing virus replication.

The key questions to be answered in the next few years are:

1. Are there, in fact, any viruses that utilize degradative autophagy to promote their replication?

2. How are autophagy-specific proteins usurped by viruses, and why? 3. Are viral infections initiating the basal-or stress-induced autophagic pathways? 4. Is blocking cargo loading essential for virus production? 5. And finally, how do the released virus-containing vesicles, derived from autophagosome inner membranes, fuse with cells to initiate new infections?

Do-Hyung Kim

Signaling

Studies on the molecular mechanisms of macroautophagy/ autophagy have been greatly advanced over the past decade with discoveries of key regulatory molecules and their signaling events. Despite the progress, we still cannot clearly explain the mechanisms that drive the dynamic membrane rearrangement during autophagy. It remains vague how post-translational modifications, interactions, and translocations of autophagy regulators are coordinated to trigger the nucleation, expansion, and closure of the autophagic membrane during autophagosome formation. To obtain clearer explanations of the long-sought-after mechanisms, we may need to decipher the coordinate regulatory relations that govern the spatiotemporal interplay between protein and lipid modifying enzymes, scaffolding proteins, membrane-binding proteins, and phospholipids. Important directions of study may also involve clarifying how autophagy is regulated in different cellular contexts via crosstalk with other signaling pathways, especially the pathways that regulate growth, death, and metabolism. The signaling mechanisms distinct between the canonical and the non-canonical autophagy pathways also remain as a key missing gap.

Daniel J. Klionsky

Other topics

There are currently 43 identified fungal ATG genes, with at least 3 more not yet published. In many cases, we still do not fully understand the function of those proteins or their mechanism of action. Further advances will likely involve additional structural studies, in particular of subcomplexes, and looking at endogenous proteins. Regulation continues to be an important area of study. One issue is that autophagy plays an essential role in many aspects of cell physiology. Thus, it will not be possible to completely shut off the process in terms of modulating autophagy for therapeutic purposes; rather, controlling the regulatory network offers a more nuanced approach to fine tuning autophagy activity.

Along these lines, a major challenge remains the demonstration that it is possible to manipulate autophagy for the purpose of improving human health. Autophagy has been implicated in a very wide array of diseases and in aging, but no one has yet shown that it is practical to modulate autophagy activity in a living person to prevent or ameliorate these conditions.

Nicholas Ktistakis

Membrane biogenesis and trafficking

One key question that is worth addressing concerns the source membrane used during selective macroautophagy/autophagy. I think there is a possibility that this may turn out to be distinct from nonselective autophagy, either because the amount of targeted cargo is limited and therefore the need for lipids to be incorporated into the forming autophagosome is not so large, or because the targeted cargo may also provide a membrane source if it happens to be itself a membrane-bound organelle (mitochondria, ER, lysosomes, etc.). However, it is also possible that a source membrane for all of these selective autophagy pathways is the ER, with ATG2 and ATG9 providing the route for lipid transfer and incorporation as is mostly the case for nonselective autophagy. A second key question concerns the role of liquid-liquid phase separated regions during autophagy induction. So far, SQSTM1 and RB1CC1/FIP200 have been found in such regions but it is likely that the list will expand. Why is it necessary to have such regions, what do they contain and what do they exclude, and how do they integrate with the downstream steps of autophagosome formation, induction and nucleation?

Marta Margeta

Neuroscience

In 2006, a couple of seminal papers conclusively demonstrated that disruption of basal macroautophagy/autophagy in the brain leads to neurodegeneration and accumulation of protein aggregates. In 16 years since that important discovery, there has been a lot of work to elucidate the role of autophagy impairment in the pathogenesis of neurodegenerative diseases and other neurological and psychiatric disorders, and much progress has been made across the board; however, many important questions remain unanswered. Here are some of them:

1. Both CNS and PNS are complex mixtures of cell types, with neurons (and their axons) only a single, albeit very important, cellular component; however, most autophagy research to date has been highly neurocentric. We need to learn a lot more about the physiological role and regulation of autophagy in all types of glial cells (astrocytes, oligodendrocytes, and Schwann cells in addition to microglia) as well as pericytes, which together with astrocytes and endothelial cells form the blood-brain and bloodnerve barriers. We also need to broadly investigate the role of non-neuronal autophagy in the pathogenesis of various CNS and PNS disease processes.

2. Recent work has demonstrated that autophagy is important for synapse maintenance, but a lot remains to be learned in this key domain of neuroscience. To what extent is autophagy regulated by neuronal activity, and is this regulation limited to neurons or does it also involve non-neuronal cell types? What is the exact role of autophagy in synaptic plasticity and synaptic pruning? How do autophagic defects in these synaptic processes contribute to developmental and psychiatric disorders such as autism and schizophrenia?

3. How are autophagic mechanisms altered across the lifespan? There has been a lot of interest in how autophagy changes with aging, but due to technical limitations the advances in this field have been relatively modest; the pace of discovery should accelerate now that better in vivo autophagy monitoring tools have become available. We also need to establish whether autophagy is altered at other key developmental time points (such as early postnatal development, puberty, and adolescence) when the brain undergoes major restructuring and is particularly vulnerable to emergence of various neuropsychiatric disorders. What is the exact role of autophagy during different stages of neurodevelopment? 4. A key but poorly understood facet of many CNS diseases is that different neuronal populations are not equally susceptible to injurious stimuli; even in the setting of a relatively well-understood external insult, injury is typically restricted to just a few neuronal subtypes, and the patterns of selective neuronal vulnerability depend on the specific disease. (For example, pyramidal neurons in the CA1 sector of the hippocampus and Purkinje neurons in the cerebellar cortex are particularly susceptible to hypoxic/ischemic injury, while dopaminergic neurons of the substantia nigra are highly vulnerable to mitochondrial dysfunction.) While the mechanisms underlying selective neuronal vulnerability are almost certainly multifactorial as well as disease dependent, there is emerging evidence to suggest that differences in either baseline or induced autophagy are partly responsible. For example, it was shown that the ischemia-resistant neurons in the CA3 sector of the hippocampus induce autophagy in response to an ischemic insult, while the ischemia-sensitive neurons in the CA1 sector do not. Similarly, global deficiency of many a key autophagy protein selectively damages cerebellar Purkinje neurons, but it is not really understood how or why. These studies have only started to scratch a surface of this important scientific problem; we need to learn a lot more about differences in autophagy regulation across different neuronal populations, and the effects of these differences on selective vulnerability to different types of injury. 5. Last, but definitely not the least, the role of autophagy impairment in the pathogenesis of adultonset neurodegenerative diseases needs to be further investigated, with Alzheimer disease (AD) emerging as a particularly complicated piece of the puzzle. On the one hand, brains of human AD patients show marked accumulation of autophagic vacuoles, reminiscent of autophagic buildup seen in genetic and toxic autophagic vacuolar myopathies. In addition, it was recently shown that a rise in autolysosomal pH is associated with Aβ accumulation within de-acidified neuronal autolysosomes in several AD mouse models. On the other hand, human genetic evidence supporting the role of autophagy dysfunction in the pathogenesis of AD is relatively weak: while genetic variants in a few proteins involved in the late stages of autophagy (such as BIN1 and PLD3 [phopholipase D family member 3]) raise the risk of late-onset AD, Mendelian disorders that affect the autophagylysosome degradation pathway have been linked to several other adult-onset neurodegenerative diseases (Parkinson disease, amyotrophic lateral sclerosis, and/or frontotemporal dementia) but not to AD.

Moreover, genetic alterations that cause human disease by altering lysosomal acidification (such as mutations in VMA21, an essential assembly chaperone of the vacuolar ATPase, which cause autophagic vacuolar myopathy) are not known to increase the risk of either early or late-onset AD, although there is some evidence that PSEN1 (presenilin 1) mutations (which cause early-onset AD) raise lysosomal pH through their effect on V-ATPase assembly. These discrepancies likely reflect the complexity of the CNS and many of the unanswered questions listed above and will need to be clarified before any autophagy-modifying therapeutic approaches can be considered for either AD or other neurodegenerative diseases.

Christian Münz

Immunology

The molecular machinery of macroautophagy/autophagy is involved in many aspects of immune responses, including elimination of intracellular pathogens, limiting inflammation, antigen processing for MHC presentation and survival of memory immune cells. Challenges in this field are dissecting which of these functions are fulfilled by canonical autophagy versus non-canonical pathways that utilize components of the autophagy machinery, such as LC3-associated phagocytosis and ATGsupported exocytosis. Furthermore, the autophagy machinery plays both beneficial and detrimental roles for different arms and stages of immune responses, such as restricting MHC class I but favoring MHC class II antigen presentation and restricting inflammation but favoring intracellular pathogen degradation. Therefore, it will be important to identify interventions and the conditions under which to apply them to augment immunity to infectious disease agents and tumors, as well as to curb autoimmunity and other immune pathologies.

Morten Petersen

Plants

A key question is to address how tissue-specific macroautophagy/autophagy contributes to plant development, and biotic and abiotic stress tolerance, and how this is coordinated at the organismal level. Autophagy has been implicated in many aspects of plant development and stress tolerance; however, we have mostly based our knowledge upon the characterization of autophagy-deficient plants. Perhaps not surprisingly, homeostatic perturbations due to the life-long loss of autophagic activity can obscure the role of this process in many aspects of plant development. Thus, conditional and/or tissue-specific knockout mutants may help to discriminate better between direct and indirect consequences of autophagic deficiencies.

We also do not know much about how different hormonal signals trigger autophagy. Is it via a common denominator or via specific components from separate signaling pathways? Another key question is therefore how do different signaling pathways employ autophagy to mediate both temporary and somatic reprogramming?

In addition to these two questions, we should also ask how to modulate autophagic activity to improve plant fitness. For example, as observed in animals, increased autophagic activity also improves plant performance. During the age of climate change autophagic activity thus represents an obvious target for agronomic intervention.

Junichi Sadoshima

Cardiovascular

Publications in the cardiovascular sector should clearly demonstrate novel molecular mechanisms and their functional significance in cardiovascular pathophysiology. In particular, addressing the cell typespecific roles of macroautophagy/autophagy in mediating the pathophysiology of the cardiovascular system is becoming increasingly important. The heart and blood vessels consist of many cell types, including cardiomyocytes, smooth muscle cells, endothelial cells, fibroblasts, inflammatory cells, etc., and the roles of autophagy and its underlying mechanisms in each cell type are quite diverse. In addition, autophagy in a particular cell type both affects the survival and death of the cell itself and influences organ physiology through autocrine/paracrine and inflammatory mechanisms [START_REF] Nussenzweig | The role of autophagy in vascular biology[END_REF]. A knowledge gap exists with regard to how autophagy and mitophagy in one cell type affect the function of other cell types. Another area of increasing importance is the study of lifestyle and dietary interventions. Investigating the involvement of autophagy and mitophagy in their salutary actions in the cardiovascular system and the underlying molecular mechanisms responsible for their effects is of great interest [START_REF] Rabinovich-Nikitin | Mitochondrial autophagy and cell survival is regulated by the circadian Clock gene in cardiac myocytes during ischemic stress[END_REF]. Finally, there is still a need for more studies addressing how basic science findings are translated to the bedside.

Isabelle Vergne

Bacterial infection

Over the past two decades, the key role of host macroautophagy/autophagy-related pathways in bacterial infection has clearly been demonstrated for a wide array of bacterial species [START_REF] Keller | Autophagy and microbial pathogenesis[END_REF]. In some cases, these pathways allow the elimination of the bacteria, whereas in others, they are manipulated by the pathogens to thrive inside the host. Although several cellular mechanisms, bacterial and host factors have been identified and characterized, our current understanding of the interplay between pathogenic bacteria and autophagy remains largely incomplete. Here, I would like to highlight two outstanding questions in the field of autophagy and bacterial infection:

1. Besides the bacteria itself, what is the full spectrum of autophagic molecular cargos during bacterial infections? Mostly hypothesis-driven approaches have hitherto been applied to identify such cargos.

A systematic approach such as proximity proteomics of autophagy receptors and LC3 proteins, combined with organelle enrichment [START_REF] Zellner | Systematically defining selective autophagy receptor-specific cargo using autophagosome content profiling[END_REF], could bring important insights into the identity of these cargos and, thus, could potentially lead to the discovery of a new function(s) of autophagy during bacterial infection.

2. Do autophagy-related pathways play a role in bacterial infection in humans? The vast majority of studies have been performed, in vitro, in 2D human cellular models. Animal models of infection, including mouse and zebrafish, have been and continue to be instrumental in visualizing autophagy and exploring its role in vivo; however, only conserved molecular mechanisms and infection processes can be investigated. In the future, the combination of patients and human 3D organoids studies will definitely improve our understanding of autophagy in bacterial infection and its relevance as a target for host-directed therapy [START_REF] Lopez-Jimenez | Emerging technologies and infection models in cellular microbiology[END_REF].

Disclosure statement

No potential conflict of interest was reported by the authors.

Funding

The authors reported there is no funding associated with the work featured in this article.


Jean Bertoin

Marie Curie

THE STRUCTURE OF THE ALLELIC PARTITION OF THE TOTAL POPULATION FOR GALTON-WATSON PROCESSES WITH NEUTRAL MUTATIONS

Keywords: AMS 2000 subject classifications. 60J80, 60J10 Branching process, infinite alleles model, allelic partition, ballot theorem

We consider a (sub-)critical Galton-Watson process with neutral mutations (infinite alleles model), and decompose the entire population into clusters of individuals carrying the same allele. We specify the law of this allelic partition in terms of the distribution of the number of clone-children and the number of mutant-children of a typical individual. The approach combines an extension of Harris representation of Galton-Watson processes and a version of the ballot theorem. Some limit theorems related to the distribution of the allelic partition are also given.

1. Introduction. We consider a Galton-Watson process, that is, a population model with asexual reproduction such that at every generation, each individual gives birth to a random number of children according to a fixed distribution and independently of the other individuals in the population. We are interested in the situation where a child can be either a clone, that is, of the same type (or allele) as its parent, or a mutant, that is, of a new type. We stress that each mutant has a distinct type and in turn gives birth to clones of itself and to new mutants according to the same statistical law as its parent, even though it bears a different allele. In other words, we are working with an infinite alleles model where mutations are neutral for the population dynamics. We might as well think of a spatial population model in which children either occupy the same location as their parents or migrate to new places and start growing colonies on their own. This quite basic framework has been often considered in the literature (see, e.g., [START_REF] Aldous | Tree-valued Markov chains derived from Galton-Watson processes[END_REF][START_REF] Crump | The dispersion of a neutral allele considered as a branching process[END_REF][START_REF] Griffiths | An infinite-alleles version of the simple branching process[END_REF][START_REF] Lyons | Probability on trees and networks[END_REF][START_REF] Nerman | Branching processes and neutral mutations[END_REF][START_REF] Taïb | Branching Processes and Neutral Evolution[END_REF]); we also refer to [START_REF] Abraham | Williams' decomposition of the Lévy continuous random tree and simultaneous extinction probability for populations with neutral mutations[END_REF][START_REF] Aldous | The standard additive coalescent[END_REF][START_REF] Aldous | Inhomogeneous continuum random trees and the entrance boundary of the additive coalescent[END_REF][START_REF] Lambert | Spine decompositions and allelic partitions of splitting trees[END_REF][START_REF] Liggett | A contact process with mutations on a tree[END_REF][START_REF] Schinazi | Spatial and nonspatial stochastic models for immune response. Markov Process[END_REF] for interesting variations (these references are of course far from being exhaustive). Note also that Galton-Watson processes with mutations can be viewed as a special instance of multitype branching processes (see Chapter V in Athreya and Ney [START_REF] Athreya | Branching Processes[END_REF] or Chapter 7 in Kimmel and Axelrod [START_REF] Kimmel | Branching Processes in Biology[END_REF]).

We are interested in the partition of the population into clusters of individuals having the same allele, which will be referred to as the allelic partition. Statistics of the allelic partition of a random population model with neutral mutations have been first determined in a fundamental work of Ewens [START_REF] Ewens | The sampling theory of selectively neutral alleles[END_REF] for the Wright-Fisher model (more precisely this concerns the partition of the population at a fixed generation). Kingman [START_REF] Kingman | Mathematics of Genetic Diversity[END_REF] provided a deep analysis of this framework, in connection with the celebrated coalescent process that depicts the genealogy of the Wright-Fisher model. We refer to [START_REF] Basdevant | Asymptotics of the allele frequency spectrum associated with the Bolthausen-Sznitman coalescent[END_REF][START_REF] Berestycki | Beta-coalescents and continuous stable random trees[END_REF][START_REF] Delmas | Asymptotic results on the length of coalescent trees[END_REF][START_REF] Dong | Exchangeable partitions derived from Markovian coalescents[END_REF][START_REF] Möhle | On sampling distributions for coalescent processes with simultaneous multiple collisions[END_REF] for some recent developments in this area which involve some related population models with fixed generational size and certain exchangeable coalescents.

The main purpose of the present work is to describe explicitly the structure of the allelic partition of the entire population for Galton-Watson processes with neutral mutations. We will always assume that the Galton-Watson process is critical or subcritical, so the descent of any individual becomes eventually extinct, and in particular the allelic clusters are finite a.s. We suppose that every ancestor (i.e., individual in the initial population) bears a different allele; it is convenient to view each ancestor as a mutant of the zeroth kind. We then call mutant of the first kind a mutant-child of an individual of the allelic cluster of an ancestor, and the set of all its clones (including that mutant) a cluster of the first kind. By iteration, we define mutants and clusters of the kth kind for any integer k ≥ 0.

In order to describe the statistics of the allelic partition, we distinguish an ancestor which will then be referred to as Eve, and focus on its descent. The set of all individuals bearing the same allele as Eve is called the Eve cluster. The Eve cluster has obviously the genealogical structure of a Galton-Watson tree with reproduction law given by the distribution of the number of clonechildren of a typical individual. Informally, the branching property indicates that the same holds for the other clusters of the allelic partition. Further, it should be intuitively clear that the process which counts the number of clusters of the kth kind for k ≥ 0 is again a Galton-Watson process whose reproduction law is given by the distribution of the number of mutants of the first kind; this phenomenon has already been pointed at in the work of Taïb [START_REF] Taïb | Branching Processes and Neutral Evolution[END_REF]. That is to say that, in some loose sense the allelic partition inherits branching structures from the initial Galton-Watson process. Of course, these formulations are only heuristic and precise statements will be given later on. We also stress that the forest structure which connects clusters of different kinds and the genealogical structure on each cluster are not independent since, typically, the number of mutants of the first kind who stem from the Eve cluster is statistically related to the size of the Eve cluster.

Our approach essentially relies on a variation of the well-known connection due to Harris [START_REF] Harris | First passage and recurrence distributions[END_REF][START_REF] Harris | The Theory of Branching Processes[END_REF] between ordinary Galton-Watson processes and sequences of i.i.d. integer-valued random variables. Specifically, we incorporate neutral mutations in Harris representation and by combination with the celebrated ballot theorem (which is another classical tool in this area as it is expounded, e.g., by Pitman; see Chapter 6 in [START_REF] Pitman | Combinatorial Stochastic Processes[END_REF]), we obtain expressions for the joint distribution of various natural variables (size of the total descent of an ancestor, number of alleles, size and number of mutant-children of an allelic cluster) in terms of the transition probabilities of the two-dimensional random walk which is generated by the numbers of clone-children and of mutant-children of a typical individual.

We also investigate some limit theorems in law; typically we show that when the numbers of clone-children and mutant-children of an individual are independent (and some further technical conditions), the sequence of the relative sizes of the allelic clusters in a typical tree has a limiting conditional distribution when the size of the tree and the number of types both tend to infinity according to some appropriate regime. The limiting distribution that arises has already appeared in the study of the standard additive coalescent by Aldous and Pitman [START_REF] Aldous | The standard additive coalescent[END_REF]. We also point at limit theorems for allelic partitions of Galton-Watson forests, where, following Duquesne and Le Gall [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF][START_REF] Duquesne | Probabilistic and fractal aspects of Lévy trees[END_REF], the limits are described in terms of certain Lévy trees. In particular, this provides an explanation to a rather striking identity between two self-similar fragmentation processes that were defined on the one hand by logging the Continuum Random Tree according to a Poisson point process along its skeleton [START_REF] Aldous | The standard additive coalescent[END_REF], and on the other hand by splitting the unit-interval at instants when the standard Brownian excursion with a negative drift reaches new infima [START_REF] Bertoin | Renewal theory for embedded regenerative sets[END_REF].

2. Allelic partitions in a Galton-Watson forest. We first develop some material and notation about Galton-Watson forests with neutral mutations, referring to Chapter 6 in Pitman [START_REF] Pitman | Combinatorial Stochastic Processes[END_REF] for background in the case without mutations.

2.1. Basic setting. Let ξ = (ξ (c) , ξ (m) )
be a pair of nonnegative integer-valued random variables which should be thought of respectively as the number of clone-children and the number of mutant-children of a typical individual. We also write

ξ (+) = ξ (c) + ξ (m)
for the total number of children, and assume throughout this work that

E(ξ (+) ) ≤ 1,
that is, we work in the critical or subcritical regime. We implicitly exclude the degenerate case when ξ (c) ≡ 0 or ξ (m) ≡ 0 and, as a consequence, the means E(ξ (c) ) and E(ξ (m) ) are always less than 1.

We write Z + and N for the sets of nonnegative integers and positive integers, respectively. A pair (g, n) ∈ Z + × N is then used to identify an individual in an infinite population model, where the first coordinate g refers to the generation and the second coordinate n to the rank of the individual of that generation (we stress that each generation consists of an infinite sequence of individuals). We assume that each individual at generation g + 1 has a unique parent at generation g. We consider a family (ξ g,n : g ∈ Z + and n ∈ N) of i.i.d. copies of ξ which we use to define the Galton-Watson process with neutral mutations. Specifically, ξ g,n = (ξ

(c) g,n , ξ (m)
g,n ) is the pair given by the number of clone-children and mutant-children of the nth individual at generation g. We may assume that the offspring of each individual is ranked, which induces a natural order at the next generation by requiring further that if (g, n) and (g, n ′ ) are two individuals at the same generation g with n < n ′ , then at generation g + 1 the children of (g, n) are all listed before those of (g, n ′ ).

2.2.

Encoding the Galton-Watson forest with mutations. Next, we enumerate as follows the individuals of the entire population (i.e., of all generations) by a variation of the well-known depth-first search algorithm that takes mutations into account. We associate to each individual a label (a, m, s), where a ∈ N is the rank of the ancestor in the initial population, m the number of mutations and s a finite sequence of positive integers which keeps track of the genealogy of the individual. Specifically, the label of the ath individual in the initial generation g = 0 is (a, 0, ∅). If an individual at the gth generation has the label (a, m, (i 1 , . . . , i g )), and if this individual has j (c) clone-children and j (m) mutant-children, then the labels assigned to its clone-children are (a, m, (i 1 , . . . , i g , 1)), . . . , (a, m, (i 1 , . . . , i g , j (c) )), whereas the labels assigned to its mutant-children are (a, m + 1, (i 1 , . . . , i g , j (c) + 1)), . . . , (a, m + 1, (i 1 , . . . , i g , j (c) + j (m) )).

Clearly, any two distinct individuals have different labels. We then introduce the (random) map which consists in ranking the individuals in the lexicographic order of their labels; see Figure 1. That is to say that ρ(i) = (g, n) if and only if the ith individual in the lexicographic order of labels corresponds to the nth individual at generation g. This procedure for enumerating the individuals will be referred to as the depth-first search algorithm with mutations. We shall also use the notation

ρ : N → Z + × N,
ξ i = ξ ρ(i) , i ∈ N,
and whenever no generation is specified, the terminology ith individual will implicitly refer to the rank of that individual induced by depth-first search with mutation, that is, the ith individual means the nth individual at generation g where ρ(i) = (g, n).

Lemma 1. (i)

The variables ξ 1 , ξ 2 , . . . are i.i.d. with the same law as ξ.

(ii) The sequence (ξ g,n : g ∈ Z + and n ∈ N) can be recovered from (ξ i : i ∈ N) a.s.

Proof. It should be plain from the definition of the depth-first search algorithm with mutations that for every i ∈ N, ρ(i + 1) is a deterministic function of ξ 1 , . . . , ξ i which takes values in (Z + × N) \ {ρ(1), . . . , ρ(i)}. Since (ξ g,n : g ∈ Z + and n ∈ N) is a sequence of i.i.d. variables with the same law as ξ, this yields the first claim by induction. The second claim follows from the fact that each individual has a finite descent a.s. [because the Galton-Watson process is (sub-)critical], which easily entails that the map ρ is bijective.

Further, it is readily seen that the inverse bijection is a function of the sequence (ξ i : i ∈ N).

Henceforth, we shall therefore encode the Galton-Watson process with neutral mutations by a sequence (ξ i : i ∈ N) of i.i.d. copies of ξ. We denote by (F i ) i∈N the natural filtration generated by this sequence.

We next briefly describe the genealogy of the Galton-Watson process as a forest of i.i.d. genealogical trees. Denote for every n ∈ N by

α n = ρ -1 (0, n), so that α 1 = 1 < α 2 < • • •
is the increasing sequence of the ranks of ancestors induced by the depth-first search algorithm with mutations. For example, α 2 = 13 in the situation described by Figure 1. The procedure for labeling individuals ensures that the descent of the ith ancestor α i corresponds to the integer interval

[α i , α i+1 [ := {α i , α i + 1, . . . , α i+1 -1}
(that is to say, if we index the population model using generations, then the descent of (0, i) is the image of [α i , α i+1 [ by the inverse bijection ρ -1 ).

We write

T i := (ξ α i -1+ℓ : 1 ≤ ℓ ≤ α i+1 -α i )
for the finite sequence of the numbers of clone-children and mutant-children of the individuals in the descent of the ith ancestor. So T i encodes (by the depth-first search algorithm with mutations) the genealogical tree of the ith ancestor, and it should be intuitively clear that the family (T i : i ∈ N) is a forest consisting in a sequence of i.i.d. genealogical trees. To give a rigorous statement, it is convenient to introduce the downward skip-free (or left-continuous) random walk

S (+) n := ξ (+) 1 + • • • + ξ (+) n -n, n ∈ Z + , (1)
and the passage times

T (+) i := inf{n ≥ 0 : S (+) n = -i}, i ∈ Z + . (2)
We stress that the T (+) i form an increasing sequence of (F n )-stopping times.

Lemma 2. There is identity

α i -1 = T (+) i-1
for every i ∈ N and, as a consequence, the sequence T 1 , . . . is i.i.d.

Proof. This formula is a close relative of the classical identity of Dwass [START_REF] Dwass | The total progeny in a branching process and a related random walk[END_REF] and would be well known if individuals were enumerated by the usual depth-first search algorithm (i.e., without taking care of mutations), see, for example, Lemma 63 in [START_REF] Pitman | Combinatorial Stochastic Processes[END_REF] or [START_REF] Gall | Random trees and applications[END_REF]. The proof in the present case is similar. Indeed the formula is obvious for i = 1, and for i = 2, we have on the one hand that

α 2 -1 = 1 + ξ (+) 1 + • • • + ξ (+) α 2 -1
by expressing the fact that the predecessor of the second ancestor found by depth-first search with mutations has a rank given by the size of the population generated by Eve, that is, Eve herself and her descendants. On the other hand, we must have 1 + ξ

(+) 1 + • • • + ξ (+)
n > n when n < α 2 -1, since otherwise the depth-first search algorithm with mutations would explore the second ancestor before having completed the exploration of the entire descent of Eve. This proves the identity for i = 2, and the general case then follows by iteration. Finally, the last claim is an immediate consequence of Lemma 1(i) and the strong Markov property.

Allelic partitions.

We can now turn our attention to defining allelic partitions. In this direction, recall that every ancestor has a different type (i.e., bears a different allele), and thus should be viewed as an initial mutant. More generally, we call mutant an individual which either belongs to the initial generation or is the mutant-child of some individual, and then write

1 = µ 1 < µ 2 < • • •
for the ranks of mutants in the depth-first search algorithm with mutations. For example, µ 2 = 6, µ 3 = 7, µ 4 = 10, µ 5 = 12 and µ 6 = α 2 = 13 in the situation depicted by Figure 1. The upshot of this algorithm is that the set of individuals that bear the same allele as the jth mutant µ j corresponds precisely to the integer interval [µ j , µ j+1 [. In this direction, it is therefore natural to introduce for every j ∈ N the jth allelic cluster

C j := (ξ µ j -1+ℓ : 1 ≤ ℓ ≤ µ j+1 -µ j ),
that is, C j is the finite sequence of the numbers of clone-children and mutantchildren of the individuals bearing the same allele as the jth mutant. The sequence (C j ) j∈N encodes the allelic partition of the entire population.

Remarks. 1. Each allelic cluster C j is naturally endowed with a structure of rooted planar tree which is induced by the Galton-Watson process. More precisely, the latter is encoded via the usual depth-first search algorithm by the sequence (ξ (c) µ j -1+ℓ : 1 ≤ ℓ ≤ µ j+1 -µ j ); in particular the jth mutant µ j is viewed as the root (i.e., ancestor) of the cluster C j . In other words, the depth-first search algorithm with mutations for the Galton-Watson process induces precisely the usual depth-first search applied to the forest of allelic clusters viewed as a sequence of planar rooted trees.

2. We also stress that the initial Galton-Watson process can be recovered from the allelic partition (C j ) j∈N . Indeed, the previous observation shows how to construct the portion of the genealogical tree corresponding to the allelic cluster generated by an initial mutant, and the latter also contains the information which is needed to identify the mutant-children of the first kind. Mutant-children of the first kind are the roots of the subtrees corresponding to the allelic clusters of the second kind, and by iteration the entire genealogical forest can be recovered.

Just as above, it is now convenient to introduce the downward skip-free random walk

S (c) n := ξ (c) 1 + • • • + ξ (c) n -n, n ∈ Z + , (3)
and the passage times

T (c) j := inf{n ≥ 0 : S (c) n = -j}, j ∈ Z + . (4)
Again, the T (c) j form an increasing sequence of (F i )-stopping times.

Lemma 3. There is identity

µ j -1 = T (c) j-1
for every j ∈ N. As a consequence, for every j ∈ N, C j is adapted to the sigma-field F T (c) j , whereas C j+1 is independent of F T (c) j and has the same distribution as C 1 . In particular the sequence of the allelic clusters

C 1 , C 2 , . . . is i.i.d.
The proof is similar to that of Lemma 2 and therefore omitted. We also introduce the number of alleles, that is, of different types, which are present in the ith tree T i :

A i := Card{j ∈ N : µ j ∈ [α i , α i+1 [};
for example, A 1 = 5 in the situation described by Figure 1. Note that there is the alternative expression

A i = 1 + α i ≤ℓ<α i+1 ξ (m) ℓ .
Corollary 1. (i) For every i ∈ Z + , we have

α i+1 = µ A 1 +•••+A i +1 ;
equivalently, there is the identity

T (+) i = T (c) A 1 +•••+A i . (ii)
The allelic partition of the tree T i , which is induced by restricting the allelic partition of the entire population to T i , is given by

(C A 1 +•••+A i-1 +ℓ : 1 ≤ ℓ ≤ A i ).
As a consequence, the sequence of the allelic partitions of the trees T i for i ∈ N, is i.i.d.

Proof. (i)

The first identity should be obvious from the definition of the depth-first search with mutations, as A 1 + • • • + A i is the number of alleles which have been found after completing the exploration of the i first trees and the next mutant is then the (i + 1)th ancestor. The second then follows from Lemmas 2 and 3.

(ii) The first assertion is immediately seen from (i) and the definitions of the trees and of the allelic clusters. Then observe that the number A i of alleles in the tree T i is a function of that tree, and so is the allelic partition. The second assertion thus derives from Lemma 2.

It may be interesting to point out that (T (+) i

, i ≥ 0) and (T (c) j , j ≥ 0) are both increasing random walks. The range

R (+) := {T (+) i : i ≥ 0}
is the set of predecessors of ancestors (in the depth-first search algorithm with mutations), whereas R (c) := {T (c) j : j ≥ 0} corresponds to predecessors of mutants. These are two regenerative subsets of Z + , in the sense that each can be viewed as the set of renewal epochs of some recurrent event (cf. Feller [START_REF] Feller | An Introduction to Probability Theory and Its Applications I[END_REF][START_REF] Feller | An Introduction to Probability Theory and Its Applications II[END_REF]). Observe that both yield a partition of the set of positive integers into disjoint intervals:

N = i≥1 ]T (+) i-1 , T (+) i ] = j≥1 ]T (c) j-1 , T (c) j ],
that correspond respectively to the trees in the Galton-Watson forest and to the allelic clusters. By Corollary 1(i), there is the embedding

R (+) ⊆ R (c)
and more precisely, this embedding is compatible with regeneration, in the sense that for every k ∈ Z + , conditionally on k ∈ R (+) , the shifted sets

R (+) • θ k := {i ≥ 0 : k + i ∈ R (+) } and R (c) • θ k := {j ≥ 0 : k + j ∈ R (c) } are
independent of the sigma-field F k generated by (ξ 1 , . . . , ξ k ) and their joint law is the same as that of (R (+) , R (c) ). We refer to [START_REF] Bertoin | Renewal theory for embedded regenerative sets[END_REF] for applications of this notion. Roughly speaking, this implies that the allelic split of each interval ]T

(+) i-1 , T (+) i ] produces smaller intervals ]T (c) j-1 , T (c)
j ] in a random way that only depends on the length

T (+) i -T (+)
i-1 (i.e., the size of T i ), independently of its location and of the other integer intervals. This can be thought of as a fragmentation property (see [START_REF] Bertoin | Random Fragmentation and Coagulation Processes[END_REF]) for the sizes of the trees.

2.4. Allelic trees and forest. In order to analyze the structure of allelic partitions, we introduce some related notions. The genealogy of the population model naturally induces a structure of forest on the set of different alleles. More precisely, we enumerate this set by declaring that the jth allele is that of the jth cluster C j , and define a planar graph on the set of alleles (which is thus identified as N) by drawing an edge between two integers j < k if and only if the parent of the kth mutant µ k is an individual of the jth allelic cluster C j . This graph is clearly a forest (i.e., it contains no cycles), which we call the allelic forest, and more precisely the ith allelic tree is that induced by the mutant descent of the ith ancestor α i . In other words, the ith allelic tree is the genealogical tree of the different alleles present in T i . In particular, the sequence of allelic trees is i.i.d. and their sizes are given by (A i , i ∈ N).

Recall that the breadth-first search in a forest consists in enumerating individuals in the lexicographic order of their labels, where the label of the nth individual at generation g is now given by the triplet (a, g, n), with a the rank of the ancestor at the initial generation. After a (short) moment of thought, we see that the definition of depth-first search with mutations for the Galton-Watson process ensures that the labeling of alleles by integers agrees with breadth-first search on the allelic forest, in the sense that the jth allele is found at the jth step of the breadth-first search on the allelic forest.

For every j ∈ N, we consider the number of new mutants who are generated by the jth allelic cluster, viz.

M j := µ j ≤ℓ<µ j+1 ξ (m) ℓ .
For instance, we have M 1 = 3, M 4 = 1 and M 2 = M 3 = M 5 = 0 in the situation depicted by Figures 1 and2. The allelic forest is thus encoded by breadth-first search via the sequence (M j , j ∈ N).

Lemma 4. The sequence (M j , j ∈ N) is i.i.d., and therefore the allelic forest is a Galton-Watson forest with reproduction law the distribution of M 1 . As a consequence, the size A 1 of the first allelic tree is given by the identity

A 1 = min{j ≥ 1 : M 1 + • • • + M j = j -1}, showing that A 1 is an (F T (c) j )-stopping time.
Proof. Recall from Lemma 3 that the sequence C 1 , C 2 , . . . of the allelic clusters is i.i.d. Clearly, each variable M j only depends on C j , which entails our first claim. The second follows from the well-known fact that breadthfirst search induces a bijective transformation between the distributions of (sub-)critical Galton-Watson forests and those of i.i.d. sequences of integervalued variables with mean less than or equal to one (see, e.g., Section 6.2 in [START_REF] Pitman | Combinatorial Stochastic Processes[END_REF]).

Finally, the identity for the number A 1 of alleles present in the tree T 1 follows from the preceding observations and again a variation of the celebrated formula of Dwass [START_REF] Dwass | The total progeny in a branching process and a related random walk[END_REF] (see, e.g., Lemma 2 in the present work), as plainly, A 1 coincides with the total size of the first tree in the allelic forest.

3. Some applications of the ballot theorem. We start by stating a version of the classical ballot theorem that will be used in this section; see [START_REF] Takács | Combinatorial Methods in the Theory of Stochastic Processes[END_REF]. Let (X 1 , . . . , X n ) be an n-tuple of random variables with values in some space E, which is cyclically exchangeable, in the sense that for every i ∈ N, there is the identity in law (X 1 , . . . , X n ) L = (X i+1 , . . . , X i+n ), where we agree that addition of indices is taken modulo n. Consider a function Lemma 5 (Ballot theorem). Under the assumptions above, the probability that the process of the partial sums of the sequence f (X 1 ), . . . , f (X n ) remains above -k until the n-step is

f : E → {-1, 0, 1, 2, . . .}
P min j ≥ 1 : j i=1 f (X i ) = -k = n = k/n.
3.1. Distribution of the allelic tree. We have now introduced all the tools which are needed for describing some statistics of the allelic partition of a Galton-Watson tree with neutral mutations. We only need one more notation. We write

π k,ℓ = P(ξ (c) = k, ξ (m) = ℓ), k, ℓ ∈ Z + , (5)
for the probability function of the reproduction law of the Galton-Watson process with mutations. For every integer n ≥ 1, we also write π * n for the nth convolution product of that law, that is,

π * n k,ℓ = P(ξ (c) 1 + • • • + ξ (c) n = k, ξ (m) 1 + • • • + ξ (m) n = ℓ).
Example. Suppose that the dynamics of the population can be described as follows. We start from a usual Galton-Watson process with reproduction law on Z + , say ̺, and assume that at each step mutations affect each child with probability p ∈ ]0, 1[, independently of the other children. In other words, the allelic forest is obtained by pruning or percolation on the genealogical forest of the Galton-Watson process, cutting each edge with probability p and independently of the other edges. See, for example, Aldous and Pitman [START_REF] Aldous | Tree-valued Markov chains derived from Galton-Watson processes[END_REF] or Chapter 4 in Lyons and Peres [START_REF] Lyons | Probability on trees and networks[END_REF]. Analytically, this means that if ξ is a random variable with law ̺, then the conditional distribution of (ξ (c) , ξ (m) ) given ξ = k is that of (k -B(k, p), B(k, p)), where B(k, p) denotes a binomial variable with parameters k and p. In this situation, it is easily seen that

π * n k,ℓ = k + ℓ k (1 -p) k p ℓ ̺ * n k+ℓ (6)
with ̺ * n denoting the nth convolution power of ̺. This expression is entirely explicit when ̺ is, for example, the Poisson, or binomial or geometric, distribution as in those cases, there are known formulas for ̺ * n . Of course, there are other natural examples in which the two-dimensional probability function π * n can be expressed in terms of simpler one-dimensional probability functions, for instance, when ξ (c) and ξ (m) are assumed to be independent or when ξ (c) = βξ and ξ (m) = (1 -β)ξ where β stands for a Bernoulli variable which is independent of ξ.

Corollary 1 enables us to restrict our attention to the allelic partition of the tree generated by a typical ancestor, say for simplicity, Eve. Recall that

T (+) 1
denotes the size of the genealogical tree T 1 of Eve, that 1 is the number of alleles found in T 1 and that the jth allelic cluster C j generates M j mutantchildren. Further, we know from Lemma 4 that the first allelic tree is encoded by breadth-first search via the finite sequence (M j , 1 ≤ j ≤ A 1 ). The latter only retains partial information about the structure of the allelic partition of T 1 , and thus it is natural to enrich it by considering more generally the sequence of pairs ((|C j |, M j ), 1 ≤ j ≤ A 1 ), where

|C j | := µ j+1 -µ j
denotes the size of the jth allelic cluster, that is, the number of individuals having the jth type. In other words, we enrich the allelic tree by assigning to each allele the size of the corresponding allelic cluster. We may now state our main result, which can be viewed as a generalization of a celebrated identity due to Dwass [START_REF] Dwass | The total progeny in a branching process and a related random walk[END_REF].

Theorem 1. (i)

The joint law of the size of T 1 and its number of alleles is given by

P(T (+) 1 = n, A 1 = k) = 1 n π * n n-k,k-1 , 1 ≤ k ≤ n.
(ii) The joint law of the size of the Eve cluster and the number of its mutant-children is given by

P(|C 1 | = n, M 1 = ℓ) = 1 n π * n n-1,ℓ , n ≥ 1 and ℓ ≥ 0. (iii) For every integers k ≥ 1, n 1 , . . . , n k ≥ 1 and ℓ 1 , . . . , ℓ k ≥ 0 such that j i=1 ℓ i > j -1 whenever 1 ≤ j < k,
we have

P(|C 1 | = n 1 , M 1 = ℓ 1 , . . . , |C k | = n k , M k = ℓ k ) = k i=1 1 n i π * n i n i -1,ℓ i .
Remarks. 1. Restricting our attention in part (iii) to sequences ℓ 1 , . . . , ℓ k ≥ 0 with inf j ≥ 1 :

j i=1 ℓ i = j -1 = k,
we stress that the statement describes the law of the entire allelic tree.

2. In particular, the law of the number A 1 of alleles is given by

P(A 1 = k) = ∞ n=1 n -1 π * n n-k,k-1 , k ≥ 0.
It may be interesting to point out that there is also the formula

P(A 1 = k) = 1 k ν * k k-1 ,
where

ν ℓ = P(M 1 = ℓ) = ∞ n=1 n -1 π * n n-1,ℓ
and ν * k the kth convolution power of ν. Indeed, this alternative formulation is seen from Lemma 4 and Dwass formula [START_REF] Dwass | The total progeny in a branching process and a related random walk[END_REF].

Proof. Recall that (ξ

(c) 1 ξ (m)
1 ), . . . , (ξ

(c) n , ξ (m)
n ) is a sequence of n i.i.d. copies of (ξ (c) , ξ (m) ) and consider the partial sums of coordinates

Σ (c) j = j i=1 ξ (c) i , Σ (m) j = j i=1 ξ (m) i and Σ j = Σ (c) j + Σ (m) j .
Introduce for every 1 ≤ k ≤ n the event

Λ n-k,k-1 = {Σ (c) n = n -k, Σ (m) n = k -1} = {Σ n = n -1, Σ (m) n = k -1}
and observe that the sequence (ξ

(c) 1 , ξ (m)
1 ), . . . , (ξ

(c) n , ξ (m)
n ) is (cyclically) exchangeable conditionally on Λ n-k,k-1 . Further, we have by definition that

P(Λ n-k,k-1 ) = π * n n-k,k-1 .
Plainly, there is the identity

{T (+) 1 = n, A 1 = k} = Λ n-k,k-1 ∩ {min{j ≥ 1 : Σ j = j -1} = n} as, according to Lemma 2, min{j ≥ 1 : Σ j = j -1} = min{j ≥ 1 : S (+) j = -1} = T (+) 1 .
By the ballot theorem [take f (x (c) , x (m) ) = x (c) + x (m) -1 in Lemma 5], we have

P(min{j ≥ 1 : Σ j = j -1} = n | Λ n-k,k-1 ) = 1/n, which yields (i).
The proof of (ii) is similar, observing that

{|C 1 | = n, M 1 = ℓ} = Λ n-1,ℓ ∩ {min{j ≥ 1 : Σ (c) j = j -1} = n}.
Finally (iii) follows by iteration from (ii) and the fact that conditionally on A 1 ≥ j + 1, the (j + 1)th allelic cluster C j+1 is independent of (C k , 1 ≤ k ≤ j) and has the same distribution as the Eve cluster C 1 (see Lemma 3).

3.2.

Conditioning on the population size and the number of alleles. In the rest of this section, we will be interested in the relative sizes of clusters in the allelic partition of the first tree T 1 , ignoring their connections. We start with a description which is essentially a variation of that in Theorem 1(iii). Recall that a random uniform cyclic permutation of {1, . . . , k}, say σ, is given by σ(i) = U + i where U is uniform on {1, . . . , k} and the addition is taken modulo k.

Corollary 2. Fix 1 ≤ k ≤ n and let σ be a random uniform cyclic permutation of {1, . . . , k} which is independent of the Galton-Watson process. Then for every collection of positive integers n 1 , . . . , n k with n 1 +• • •+n k = n, we have

P(|C σ(1) | = n 1 , . . . , |C σ(k) | = n k | T (+) 1 = n, A 1 = k) = n kπ * n n-k,k-1 k i=1 1 n i π * n i n i -1,ℓ i ,
where in the right-hand side, the sum is taken over the sequences ℓ 1 , . . . , ℓ k in Z + such that

ℓ 1 + • • • + ℓ k = k -1.
Proof. A classical application of the ballot theorem shows that the conditional distribution of (C σ(1) , . . . , C σ(k) ) given

T (+) 1 = n and A 1 = k is the same as that of (C 1 , . . . , C k ) conditioned on k i=1 |C i | = n and k i=1 M i = k -1. Then note that k i=1 |C i | = n and k i=1 M i = k -1 ⇐⇒ T (c) k = n and n i=1 ξ (m) i = k -1
and an application of the ballot theorem (much in the same way as in the proof of Theorem 1) shows that the probability of that event equals

k n π * n n-k,k-1 .
Theorem 1(ii) completes the proof.

Next, we normalize the size |C i | of each cluster by the size T (+) 1

of the total population (recall we focus on the descent of a single ancestor, namely Eve), and write

Γ 1 ≥ Γ 2 ≥ • • • ≥ Γ A 1
for the sequence which is obtained by ranking the ratios

|C i |/T (+) 1
in the decreasing order. So Γ = (Γ 1 , . . . , Γ A 1 ) is a proper partition of the unit mass, in the sense that it is given by a ranked sequence of positive real numbers with sum 1. The space of mass partitions (possibly with infinitely many strictly positive terms and sum less than 1) is endowed with the supremum distance, which yields a compact metric space; see Section 2.1 in [START_REF] Bertoin | Random Fragmentation and Coagulation Processes[END_REF] for details.

Our purpose now is to investigate the asymptotic behavior of the random mass partition Γ, under the conditional probability given the size T (+) 1

= n of the tree T 1 and the number A 1 = k of alleles, when n, k → ∞. We shall show that, under appropriate hypotheses, one can establish convergence in distribution, where the limit can be described as follows. For some fixed parameter b > 0, consider the sequence a 1 > a 2 > • • • > 0 of the atoms ranked in the decreasing order of a Poisson point measure on ]0, ∞[ with intensity ba -3/2 da. Roughly speaking, we then get a random proper mass-partition by conditioning on ∞ i=1 a i = 1; see, for example, [START_REF] Perman | Size-biased sampling of Poisson point processes and excursions[END_REF] or Proposition 2.4 in [START_REF] Bertoin | Random Fragmentation and Coagulation Processes[END_REF] for a rigorous definition of this conditioning by a singular event.

This family of random mass-partitions has appeared previously in a remarkable work by Aldous and Pitman [START_REF] Aldous | The standard additive coalescent[END_REF], more precisely it arose by logging the Continuum Random Tree according to Poissonian cuts along its skeleton; see also [START_REF] Abraham | Poisson snake and fragmentation[END_REF][START_REF] Aldous | Inhomogeneous continuum random trees and the entrance boundary of the additive coalescent[END_REF][START_REF] Bertoin | A fragmentation process connected to Brownian motion[END_REF][START_REF] Miermont | Ordered additive coalescent and fragmentations associated to Lévy processes with no positive jumps[END_REF] for related works. In the present setting, we may interpret such cuts as mutations which induce an allelic partition. As we know from Aldous [START_REF] Aldous | The continuum random tree[END_REF] that the Continuum Random Tree can be viewed as the limit when n → ∞ of Galton-Watson trees conditioned to have total size n, the fact that the preceding random mass-partitions appear again in the framework of this work should not come as a surprise.

For the sake of simplicity, we shall focus on the case when the number of clone-children ξ (c) and the number of mutant-children ξ (m) are independent, although it seems likely that our argument should also apply to more general situations. Recall that the expected number of clone-children of a typical individual is E(ξ (c) ) < 1. We shall work under the hypothesis that by a suitable exponential tilting, this subcritical random variable can be turned into a critical one with finite variance. That is, we shall assume that there exists a real number θ > 1 such that

E(ξ (c) θ ξ (c) ) = E(θ ξ (c) ) and σ 2 θ := E((ξ (c) ) 2 θ ξ (c) )/E(θ ξ (c) ) -1 < ∞. (7)
It can be readily checked that [START_REF] Aldous | Inhomogeneous continuum random trees and the entrance boundary of the additive coalescent[END_REF] then specifies θ uniquely.

Proposition 1. Suppose that ξ (c) and ξ (m) are independent, that neither distribution is supported by a strict subgroup of Z and that [START_REF] Aldous | Inhomogeneous continuum random trees and the entrance boundary of the additive coalescent[END_REF] Remark. The special case when ξ (c) and ξ (m) are two Poisson variables, say with rates r (c) and r (m) can also be viewed as an instance of the situation where mutations affect children independently with probability p = r (m) /(r (c) + r (m) ) (cf. the example discussed before Theorem 1). More precisely the reproduction law of the standard Galton-Watson process is then Poisson with rate r (c) + r (m) . This special case has some importance, as it is well known that conditioning a Galton-Watson tree with Poisson(1) reproduction law to have a size n and then assigning to each individual a distinct label in {1, . . . , n} by uniform sampling without replacements yields the uniform distribution on the set of rooted trees with n labeled vertices.

Proof. Let P denote the probability measure which is obtained from P by exponential tilting, and more precisely, in such a way that the variables ξ (c) 1 , . . . are i.i.d. under P with law given by P(ξ (c) = j) = θ j P(ξ (c) = j)/z θ , j ∈ Z + , where z θ is the normalization factor, namely,

z θ = E(θ ξ (c) ).
As in the proof of Corollary 2, we see from an application of the ballot theorem that the conditional distribution of (nΓ 1 , . . . , nΓ A 1 ) given T

= n).

We then estimate the distribution of the size of the Eve cluster under P, which is given again according to the Dwass formula [START_REF] Dwass | The total progeny in a branching process and a related random walk[END_REF] by

P(|C 1 | = n 1 ) = 1 n 1 P(ξ (c) 1 + • • • + ξ (c) n 1 = n 1 -1) = 1 n 1 P(S (c) n 1 = -1).
Recall that, by assumption, ξ (c) is critical with variance σ 2 θ under P, so an application of Gnedenko's local central limit theorem gives

P(|C 1 | = n 1 ) ∼ 1 2πσ 2 θ n 3 1 as n 1 → ∞.
Putting the pieces together, we get that the conditional distribution of (nΓ 1 , . . . , nΓ A 1 ) given T (+) 1

= n and A 1 = k is the same as that obtained from an i.i.d. sequence Y 1 , . . . , Y k by ranking in the decreasing order and conditioning on k i=1 Y i = n, where

P(Y 1 = n 1 ) ∼ 1 2πσ 2 θ n 3 1 as n 1 → ∞.
An application of Corollary 2.2 in [START_REF] Bertoin | Random Fragmentation and Coagulation Processes[END_REF] completes the proof of our claim.

4.

Lévy forests with mutations. The purpose of this section is to point at an interpretation of a standard limit theorem involving left-continuous (i.e., downward skip-free) random walks and Lévy processes with no negative jumps, in terms of Galton-Watson and Lévy forests in the presence of neutral mutations. We first introduce some notation and hypotheses in this area, referring to the monograph by Duquesne and Le Gall [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF] for details.

For every integer n ≥ 1, let (ξ (c) (n), ξ (m) (n)) be a pair of integer-valued random variables with

E(ξ (c) (n) + ξ (m) (n)) = 1.
We consider two left-continuous random walks

S (+) (n) = (S (+) i (n) : i ∈ Z + ) and S (c) (n) = (S (c) i (n) : i ∈ Z + ), whose steps are (jointly) distributed as ξ (+) (n) := ξ (c) (n) + ξ (m) (n) -1 and ξ (c) (n) -1,
respectively. Let also X = (X t , t ∈ R + ) denote a Lévy process with no negative jumps and Laplace exponent ψ, namely,

E(exp -λX t ) = exp tψ(λ)
for every λ, t ≥ 0.

We further suppose that X does not drift to +∞, which is equivalent to ψ ′ (0+) ≥ 0, and that

∞ 1 dλ ψ(λ) < ∞.
We also need to introduce a different procedure for encoding forests by paths, which is more convenient to work with when discussing continuous limits of discrete structures. For each n ≥ 1, we write

H(n) = (H i (n), i ∈ N) for the (discrete) height function of the Galton-Watson forest (T ℓ , ℓ ∈ N).
That is, for i ≥ 0, H i (n) denotes the generation of the (i + 1)th individual found by the usual depth-first search (i.e., mutations are discarded) on the Galton-Watson forest. In the continuous setting, trees and forests can be defined for a fairly general class of Lévy processes with no negative jumps, and in turn are encoded by (continuous) height functions; cf. Chapter 1 in [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF] for precise definitions and further references.

The key hypothesis in this setting is the existence of a nondecreasing sequence of positive integers (γ n , n ∈ N) converging to ∞ and such that

lim n→∞ n -1 S (+) nγn (n) = X 1 in law; (8)
we also assume that the technical condition (2.27) in [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF] is fulfilled. Then the rescaled height function (γ -1 n H [tnγn] (n) : t ≥ 0) converges in distribution, in the sense of weak convergence on Skorohod space D(R + , R + ) as n → ∞ toward the height process (H t : t ≥ 0) which is constructed from the Lévy process X = (X t , t ≥ 0); see Theorem 2.3.1 in [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF].

Similarly, we write

H (c) (n) = (H (c) i (n), i ∈ N)
for the height function of the Galton-Watson forest (C j , j ∈ N), where each allelic cluster C j is endowed with the genealogical tree structure induced by the population model (see Remark, item 1 in Section 2.3). Proposition 2. Suppose that the preceding assumptions hold, and also that lim Remark. More recently, Duquesne and Le Gall [START_REF] Duquesne | Probabilistic and fractal aspects of Lévy trees[END_REF] (see also the survey [START_REF] Gall | Random trees and applications[END_REF]) have developed the framework when Lévy trees are viewed as random variables with values in the space of real trees, endowed with the Gromov-Hausdorff distance. Proposition 2 can also be restated in this setting.

Proof of Proposition 2. The assumption [START_REF] Athreya | Branching Processes[END_REF] ensures the convergence in distribution (n -1 S (+)

[tnγn] (n) : t ≥ 0) =⇒ (X t : t ≥ 0), see Theorem 2.1.1 in [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF] and (2.3) there. On the other hand, by a routine argument based on martingales, the assumption (9) entails that lim [tnγn] (n) : t ≥ 0) =⇒ (X t -dt : t ≥ 0) follows. Recall that depth-first search with mutations on the initial forest yields the usual depth-first search for the forest of allelic clusters (cf. Remark, item 1 in Section 2.3). We can then complete the proof as in Theorem 2.3.1 in [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF].

We now conclude this work by discussing a natural example. Specifically, we suppose that the distribution of ξ (c) (n) + ξ (m) (n) = ξ(n) := ξ is the same for all n. For the sake of simplicity, we assume also that E(ξ) = 1 and Var(ξ) = 1. We may then take γ n = n, so by the central limit theorem, (8) holds and the Lévy process X is a standard Brownian motion. We fix an arbitrary d > 0 and consider the independent pruning model where for each integer n > d, conditionally on the total number of children ξ (+) (n) := ξ (c) (n) + ξ (m) (n) = k, the number ξ (m) (n) of mutant-children of a typical individual has the binomial distribution B(k, d/n). In other words, in the nth population model, mutations affect each child with probability d/n, independently of the other children. Then (9) clearly holds. Roughly speaking, Theorem 2.3.1 of [START_REF] Duquesne | Random trees, Lévy processes and spatial branching processes[END_REF] implies in this setting that the initial Galton-Watson forest associated with the nth population model, converges in law after a suitable renormalization to the Brownian forest, whereas Proposition 2 of the present work shows that the allelic forest renormalized in the same way, converges in law to the forest generated by a Brownian motion with drift -d.

This provides an explanation to the rather intriguing relation which identifies two seemingly different fragmentation processes: the fragmentation process constructed by Aldous and Pitman [START_REF] Aldous | The standard additive coalescent[END_REF] by logging the Continuum Random Tree according to a Poisson point process on its skeleton, and the fragmentation process constructed in [START_REF] Bertoin | A fragmentation process connected to Brownian motion[END_REF] by splitting the unit interval at instants when a Brownian excursion with negative drift reaches a new infimum. It is interesting to mention that Schweinsberg [START_REF] Schweinsberg | Applications of the continuous-time ballot theorem to Brownian motion and related processes[END_REF] already pointed at several applications of the (continuous) ballot theorem in this framework. More generally, the transformation X → X (d) of Lévy processes with no negative jumps also appeared in an article by Miermont [START_REF] Miermont | Ordered additive coalescent and fragmentations associated to Lévy processes with no positive jumps[END_REF] on certain eternal additive coalescents, whereas Aldous and Pitman [START_REF] Aldous | Inhomogeneous continuum random trees and the entrance boundary of the additive coalescent[END_REF] showed that the latter arise asymptotically from independent pruning of certain sequences of birthday trees. Finally, we also refer [START_REF] Abraham | Pruning a Lévy continuum random tree[END_REF] for another interesting recent work on pruning Lévy random trees.

Fig. 1 .

 1 Fig. 1. Depth-first search with mutations on a genealogical tree. The symbols •, ♠, ♥, ♦, ♣ represent the different alleles. Left: the label (m, s) of an individual is given by the number m of mutations and the sequence s that specifies its genealogy; for the sake of simplicity, the rank a of the ancestor has been omitted. Right: the same tree with individuals ranked by the depth-first search algorithm with mutations.

Fig. 2 .

 2 Fig. 2. Allelic tree corresponding to the genealogical tree with mutations in Figure 1. The labels represent the sizes of the allelic clusters.

) 1 =

 1 holds. Fix b > 0 and let n, k → ∞ according to the regime k ∼ b √ n. Then the conditional law of Γ given that the size of the total population is T (+n and the number of alleles A 1 = k converges weakly on the space of mass-partitions to the sequence (a 1 , a 2 , . . .) of the atoms of a Poisson random measure on ]0, ∞[ with intensity b 2πσ 2 θ a 3 da, a > 0, ranked in the decreasing order and conditioned by ∞ i=1 a i = 1.

A 1 = 1

 11 k is the same as that obtained from the i.i.d. sequence |C 1 |, . . . , |C k | by ranking in the decreasing order and conditioning on k i=1 |C i | = n and k i=1 M i = k -1. Observe that the latter is equivalent to conditioning on k i=1 |C i | = n and n i=1 ξ (m) i = k -1. Further, recall from Lemma 3 that |C j | = T and hence, on this event, the variables |C 1 |, . . . , |C k | are functions of ξ (c) 1 , . . . , ξ (c) n . Thus the assumption of independence between ξ (c) and ξ (m) enables us to ignore the conditioning on n i=1 ξ (m) i = k -1. Finally, it should be clear that the exponential tilting does not affect such a conditional law, in the sense that the sequence |C 1 |, . . . , |C k | has the same distribution under P(• | T (+) 1 = n) as under P(• | T (+) 1

  n→∞ γ n E(ξ (m) (n)) = d and lim n→∞ n -1 γ n Var(ξ (m) (n)) = 0 (9)for some d ≥ 0. Then the rescaled height function(γ -1 n H (c)[tnγn] (n) : t ≥ 0) converges in distribution, in the sense of weak convergence on Skorohod space D(R + , R + ) as n → ∞ toward the height process (H (d) t : t ≥ 0), which is constructed from the Lévy process X (d) = (X (d) t := X t -dt, t ≥ 0).

  n→∞ n -1 (S (+) [tnγn] (n) -S (c) [tnγn] (n)) = dt,uniformly for t in compact intervals, in L 2 (P). The convergence in distribution (n -1 S (c)

J. BERTOIN

Acknowledgment. would like to thank two anonymous referees for their careful check of this work.