Jean-Christophe Gelly
email: jean-christophe.gelly@univ-paris-diderot.fr

Alexandre G De Brevern
email: alexandre.debrevern@univ-paris-diderot.fr

Dr G De Alexandre

Protein Peeling webserver

Analysis of the architecture and organization of protein structures is a major challenge to better understand protein flexibility, folding, functions and interactions with their partners and to design new drugs.

Protein structures are often described as series of α-helices and β-sheets, or at a higher level as an arrangement of protein domains. Due to the lack of an intermediate vision which could give a good understanding and description of protein structure architecture, we have proposed a novel intermediate view, the Protein Units (PUs). They are novel level of protein structure description between secondary structures and domains. A PU is defined as a compact sub-region of the 3D structure corresponding to one sequence fragment, defined by a high number of intra-PU contacts and a low number of inter-PU contacts. The methodology to obtain PUs from the protein structures is named Protein Peeling (PP). For the algorithm, the protein structures are described as a succession of Cα. The distances between Cα are translated into contact probabilities using a logistic function. Protein Peeling only uses this contact probability matrix. An optimization procedure, based on the Matthews' coefficient correlation (MCC) between contacts probability sub matrices, defines optimal cutting points that separate the region examined into two or three PUs. The process is iterated until the information necessary for two others softwares (mainly coded in C to perform most of the computation tasks and R for the analysis). Results are given both textually and graphically using JMol applet and PyMol software. The server can be accessed from http://www.dsimb.inserm.fr/dsimb_tools/peeling/. Only one equivalent on line methodology is available.

compactness of the resulting PUs reaches a given limit. An index assesses the compactness quality and relative independence of each PU.

Protein Peeling is a tool to better understand and analyze the organization of protein structures. We have developed a dedicated bioinformatic web server: Protein Peeling 2 (PP2).

Given the 3D coordinates of a protein, it proposes an automatic identification of protein units (PUs). The interface component consists of a web page (HTML) and common gateway interface (CGI). The user can set many parameters and upload a given structure in PDB file format to a perl core instance. This last component is a module that embeds all the

Introduction

The proteins are a succession of amino acids joined together by peptide bonds. They are crucial macromolecules implicated in major physiological processes and also most of the diseases. Since the elucidation of the first protein structure by Max Perutz and John Kendrew in the late 50's [START_REF] Kendrew | A three-dimensional model of the myoglobin molecule obtained by x-ray analysis[END_REF][START_REF] Perutz | Structure of haemoglobin: a three-dimensional Fourier synthesis at 5.5-A. resolution, obtained by X-ray analysis[END_REF], it was experimentally demonstrated that functional proteins adopt a three dimensional structure (3D) defined by the spatial arrangement of the atoms of its amino acids. Protein 3D structures are still resolved using Xray crystallography since the last 50 years [START_REF] Kendrew | A three-dimensional model of the myoglobin molecule obtained by x-ray analysis[END_REF]. The process by which a protein adopts this three-dimensional structure under natural condition from an initial disordered state is called folding. Native protein structures are maintained by inter-residue interactions. Anfinsen demonstrated that the amino acid sequence alone contain all the information needed to obtain a functional protein structure [START_REF] Anfinsen | The kinetics of formation of native ribonuclease during oxidation of the reduced polypeptide chain[END_REF]. Otherwise molecular mechanism responsible for this self-assembly is poorly understood and remains one of the most fundamental problems in biological sciences.

From the beginning of biochemical sciences, interesting characteristics have been determined or theoretically predicted. Thus, some amino acids favor to adopt local structures called repetitive secondary structures: α-helix and β-sheet due to their physicochemical properties. The combination of theses secondary structures elements and other non-regular form the final structure of the protein and characterize a particular and functional protein topology [START_REF] Richardson | The anatomy and taxonomy of protein structure[END_REF].

Various studies had also shown that protein structure fold can be represented into units called protein structural domains. Proteins can be constituted by one unique domain while others are combinations of many ones. Domains represent not only structural meaningful elements but also facilitate the understanding of protein architecture. Quasi structural independence is the major characteristic of domains. Sometimes theses domains had also a well defined function and were evolutionary conserved [START_REF] Ponting | The natural history of protein domains[END_REF]. One aim is to simplify analysis into more significant component based on geometric and physicochemical properties. Indeed great part of protein domains are organized around a hydrophobic core and some are able to fold independently and exhibit a well defined topology. More than one thousand different domains have been identified in structural databases, e.g., SCOP [START_REF] Murzin | SCOP: a structural classification of proteins database for the investigation of sequences and structures[END_REF], CATH [START_REF] Orengo | CATH--a hierarchic classification of protein domain structures[END_REF] or FSSP [START_REF] Holm | Dali/FSSP classification of three-dimensional protein folds[END_REF]. Defining automatic procedures for reliable domain assignment is an essential task for the generation of pertinent domain databases used for relevant scientific studies [START_REF] Heger | ADDA: a domain database with global coverage of the protein universe[END_REF]. The main idea behind these approaches is that the inter-domain interaction is weaker than the intra-domain interaction.

Despite availability of these various tools, it remains hard to describe and understand protein structures diversity with theses methodologies. A clear gap exists between an elementary view of protein structure as a succession of secondary structure elements, and a more complex view of protein structural domains. It lacks a level of description complementary between secondary structures and domains, a kind of intermediate view of structural organization and complexity. Some authors have proposed such a supplementary level, consistent with folding models. Wetlaufer was the first to examine the organization of known structures and suggested that the early stages of 3D structure formation, i.e., nucleation, occur independently in separate parts of these molecules [START_REF] Wetlaufer | Nucleation, rapid folding, and globular intrachain regions in proteins[END_REF]1981). These folding units have been proposed to fold independently during the folding process, creating structural modules which give birth to the native structure.

Protein domains identification methods. Protein structures can be seen as composed of single or multiple functional domains that can fold and function independently. Dividing a protein into domains is useful for more accurate structure and function determination and explanation of folding process. Automatic domain parsing is based on a common simple principle: inter-domain interaction is weaker than the intra-domain interaction and intradomain are strong enough to maintain stability [START_REF] Wetlaufer | Nucleation, rapid folding, and globular intrachain regions in proteins[END_REF][START_REF] Rossman | Letter: Recognition of structural domains in globular proteins[END_REF][START_REF] Richardson | The anatomy and taxonomy of protein structure[END_REF][START_REF] Wetlaufer | Folding of protein fragments[END_REF]. Domain definition is simply the result of this cutting process. Many different procedures to assign protein structural domains have been developed. DETECTIVE method is based on the idea that domains have a hydrophobic interior [START_REF] Swindells | A procedure for detecting structural domains in proteins[END_REF], while Wodak and Janin used an iterative approach based on surface areas with an iterative cleavage of the native structure [START_REF] Wodak | Location of structural domains in protein[END_REF]. Gaussian network model could also be used [START_REF] Kundu | Automatic domain decomposition of proteins by a Gaussian Network Model[END_REF]). Different algorithms to hierarchically split proteins into compact units have been proposed [START_REF] Lesk | Folding units in globular proteins[END_REF][START_REF] Wodak | Location of structural domains in protein[END_REF][START_REF] Sowdhamini | An automatic method involving cluster analysis of secondary structures for the identification of domains in proteins[END_REF][START_REF] Swindells | A procedure for detecting structural domains in proteins[END_REF][START_REF] Tsai | Hydrophobic folding units derived from dissimilar monomer structures and their interactions[END_REF][START_REF] Kundu | Automatic domain decomposition of proteins by a Gaussian Network Model[END_REF]). Their goal was to describe protein structure at different organization levels [START_REF] Taylor | Evolutionary transitions in protein fold space[END_REF]. Nonetheless, the problem of dividing a protein structure into domains is not yet solved.

PUU [START_REF] Holm | Parser for protein folding units[END_REF]) is a recursive top-down approach which uses a hypothetical model of autonomously folding units corresponding to protein domains. A hierarchical 5-level filtering process is applied during partitioning of the structure, it tries to conserve long length protein fragment, cut flexible regions and not secondary structure. DomainParser uses a topdown approach to domain decomposition implemented using a graph theoretical approach [START_REF] Xu | Protein domain decomposition using a graphtheoretic approach[END_REF][START_REF] Guo | Improving the performance of DomainParser for structural domain partition using neural network[END_REF]. Its main problem is the failure to continue successful partitioning. Protein Domain Parser [START_REF] Alexandrov | PDP: protein domain parser[END_REF] is based on the assumption that the expected number of contacts between two domains depends on their surface areas. DOMAK [START_REF] Siddiqui | Continuous and discontinuous domains: an algorithm for the automatic generation of reliable protein domain definitions[END_REF], 3Dee [START_REF] Dengler | Protein structural domains: analysis of the 3Dee domains database[END_REF][START_REF] Siddiqui | 3Dee: a database of protein structural domains[END_REF], DETECTIVE [START_REF] Swindells | A procedure for detecting structural domains in proteins[END_REF], DALI [START_REF] Holm | Dictionary of recurrent domains in protein structures[END_REF], STRUDL [START_REF] Wernisch | Identification of structural domains in proteins by a graph heuristic[END_REF], and DDOMAIN [START_REF] Zhou | DDOMAIN: Dividing structures into domains using a normalized domain-domain interaction profile[END_REF])) are build on similar approaches.

Interestingly, they are often benchmarked on a manual definition of structural domains [START_REF] Joshi | A Decade of Computing to Traverse the Labyrinth of Protein Domains[END_REF] as SCOP [START_REF] Murzin | SCOP: a structural classification of proteins database for the investigation of sequences and structures[END_REF]. The difficulty of defining automatically structural domains has been often been shown, e.g., [START_REF] Holland | Partitioning protein structures into domains: why is it so difficult?[END_REF] and no consensus could be easily found. An important point is the size of protein domains which always remains important (often more than a hundred residues) and so does not reflect protein folding early steps.

Small compact unit identification methods. Thus, many researchers have tried to determined smaller protein units which could represent earlier event of the protein folding and the smallest basic element of structure organization. The most common view was to define a hierarchically splitting of proteins into compact units [START_REF] Go | Correlation of DNA exonic regions with protein structural units in haemoglobin[END_REF][START_REF] Lesk | Folding units in globular proteins[END_REF][START_REF] Wodak | Location of structural domains in protein[END_REF][START_REF] Janin | Structural domains in proteins and their role in the dynamics of protein function[END_REF][START_REF] Sowdhamini | An automatic method involving cluster analysis of secondary structures for the identification of domains in proteins[END_REF][START_REF] Tsai | Hydrophobic folding units derived from dissimilar monomer structures and their interactions[END_REF][START_REF] Guo | Improving the performance of DomainParser for structural domain partition using neural network[END_REF][START_REF] Pugalenthi | DIAL: a web-based server for the automatic identification of structural domains in proteins[END_REF]. During the 70's Wetlaufer proposed that the early stages of 3D structure formation, i.e., nucleation, occur independently in separate parts of these molecules [START_REF] Wetlaufer | Nucleation, rapid folding, and globular intrachain regions in proteins[END_REF]1981). These folding units have been proposed to fold independently during the folding process, creating structural modules which can be assembled to give the native structure.

Go identifies basic structural unit by Cα-Cα distance map and visual inspection of protein structures [START_REF] Go | Correlation of DNA exonic regions with protein structural units in haemoglobin[END_REF]) while Janin and Wodak algorithm search along polypeptide the partitioning point generating units with the smallest surface interaction [START_REF] Janin | Structural domains in proteins and their role in the dynamics of protein function[END_REF]. Folding unit as defined by [START_REF] Lesk | Folding units in globular proteins[END_REF] are selected by a bottom-up hierarchical approach using inertial ellipsoids minimal area of small fragments.

Later many methods based on different principles have been proposed. One of the most recent methodology is DIAL [START_REF] Sowdhamini | An automatic method involving cluster analysis of secondary structures for the identification of domains in proteins[END_REF][START_REF] Pugalenthi | DIAL: a web-based server for the automatic identification of structural domains in proteins[END_REF]) and his database [START_REF] Sowdhamini | A database of globular protein structural domains: clustering of representative family members into similar folds[END_REF]. DIAL algorithm determined small compact unit by hierarchical approach based on distances between secondary structure elements.

Protein Unit and Protein Peeling. We have likewise developed a method called Protein Peeling (Gelly et al. 2006a). This algorithm dissects a protein into Protein Units (PUs). A PU is a compact sub-region of the 3D structure corresponding to one sequence fragment. The basic principle is that each PU must have a high number of intra-PU contacts, and, a low number of inter-PU contacts. Thus, organization of protein structures can be considered in a hierarchical manner: secondary structures are the smallest elements, and, Protein Units are intermediate elements leading to structural domains.

Bioinformatics methodology needs to be widely available and distributed to be useful for the scientific community. A web server (http://www.dsimb.inserm.fr/dsimb_tools/peeling/) dedicated to Protein Peeling, has been developed for this purpose. It is now the second improved version of our approach (Gelly et al. 2006b).

Methodology of Protein Peeling

Protein Peeling algorithm works from the Cα-contact matrix translated into contact probabilities. A PU (or the protein at the beginning) is associated to a protein sequence s comprised between positions [i, j], (i < j, i=1 and j=N at the beginning).The sequence is cut into two parts s 1 and s 2 associated with the positions [i, m] and [m + 1, j] respectively. The symmetric contact probability sub-matrix associated to the sequence s is shared into 3 submatrices corresponding to the sum of the contact probabilities between the residues of s 1 with itself (noted A), s 2 with itself (B), and, s 1 with s 2 (C). To assess the presence of numerous contacts within the sub-units s 1 and s 2 and a limited number of contacts between them, Matthews' coefficient correlation (MCC) is used [START_REF] Matthews | Comparison of the predicted and observed secondary structure of T4 phage lysozyme[END_REF]. The MCC measure is translated into a partition index, PI ij (m):

Thus, the quality of the splitting of the PU into two sub-units is quantified via a correlation. The complete absence of contacts between these two sub-units (i.e., C = 0) leads to a maximal value of the partition index (i.e., 1). A large presence of contacts between subunits (C > 0) induces a low PI value. The cutting process cuts in 2 or 3PUs. To characterize the compactness of PUs defined, a compaction index (CI) based on mutual information is calculated [START_REF] Etchebest | A structural alphabet for local protein structures: improved prediction methods[END_REF][START_REF] Hazout | Entropy-derived measures for assessing the accuracy of N-state prediction algorithms[END_REF])., it uses the sum of the probabilities associated with each PUs and indicates when to stop cutting, when it reaches a given threshold R (see (Gelly et al. 2006a) for more details and especially the Figure 1).

The process is recursively done. It is iterated until the compactness of the resulting PUs reaches a given limit, fixed by the user. A refinement of cutting is carried out thanks to the method of pruning which checks that PUs lately generated are compact (Gelly et al. 2006b).

2

, ( ) ( )( )

i j AB C m PI A C B C - = + +

Comparable approaches

The only comparable method is so DIAL. The approach is available at http://caps.ncbs.res.in/DIAL/. It considers small units as clusters of secondary structure elements. In a first step, α-helices and β-strands are first clustered using inter-secondary structural distances between Cα positions. In a second step, dendograms based on this distance measure are used to identify sub-domains. Their goal was to describe the different levels of protein structure organization.

Figure 1 shows the use of DIAL Web server to cut the structure of dialkylglycine decarboxylase (PDB code 1ZOB [START_REF] Berman | The Protein Data Bank[END_REF]). Figure 1a shows the website, Figure 1b is the result of cutting by DIAL of dialkylglycine decarboxylase. For this particular protein, two regions are found. They are represented on the structure on Figure 1c. It is a nice static view of the protein using molscript software [START_REF] Kraulis | MOLSCRIPT: a program to produce both detailed and schematic plots of protein structures[END_REF].

Protein Peeling web server

The flowchart representation of Protein Peeling web server is shown on Figure 2. 1983). Then, the perl module launches the protein peeling main software (done in C language for computational efficiency), that reads the clean PDB file, the secondary structure assignment and the different options. The protein peeling process is done and compactness indices are computed.

In a second step, render programs perform visualization of results. R software scripts [START_REF] Ihaka | R: a language for data analysis and graphics[END_REF] (ii) An interactive visualization is also possible through the JMol applet (JMol) which is based on a Java Virtual Machine.

The perl core generates finally a complete web page (see Figure 2, left) that summarizes all the output information. Peeling approach (PDB code 1ZOB). Figure 4b shows the dendogram obtained with default parameters; the cutting is so quite impressive. For R 2 equals to 20, a first event appears, it cuts the protein into two much misbalanced PUs (1-26 and 27-431). In a recent study [START_REF] Faure | Analysis of protein contacts into Protein Units[END_REF], we have shown that Protein Peeling can detect mobile extremities. These last have fewer constraints than the hydrophobic core of the protein and so are often considered as "mobile" (Jacob and Unger 2007). Our "mobile" extremities have been detected as PU, representing less than 20% of the size of protein which are cut early in the process of peeling and is not cut again. Half of the proteins have been detected associated to mobile extremities.

Example of protein cutting

Here, our mobile N-terminus (residue 1-26) is mainly helical which the case is for 2/3 of Ntermini. As α-helices are not conditioned by long range contacts within the sequence like βsheets; this tendency seems logical. Its CI value is low (0.26).

The second cutting event is for R 2 equals to 70, the splitting event is not a simple dual one, but three PUs are generated, with one short (27-59) and two longer PUs (60-326 and 327-431). The first one mainly composed of β-strands will not be cut again as the last one which is a bundle of 3 α-helices and 3 β-strands. They are associated to very high CI values

(2.49 and 3.75 respectively). Next cutting events cut so the central PUs into 3 PUs and at the end finally into 8 PUs.

Depending on the purpose of the research, the final number PUs and / or their lengths and contents can be different. Here some PUs are only 20 residues long and associated to low CI, e.g., PU 60-80 has null CI. It is so interesting to come back at previous cutting events. Our web server allows coming back to each cutting events. It is also always possible to change the options concerning length, R 2 values, etc.

Conclusion

The three-dimensional structure is the core of protein functions and is mainly determined by its amino acid sequence. Nonetheless, the protein folding is not completely understood [START_REF] Clark | Protein folding: Are we there yet?[END_REF]). Several models have been proposed for protein folding, e.g., the framework model [START_REF] Ptitsyn | A model of myoglobin self-organization[END_REF][START_REF] Udgaonkar | NMR evidence for an early framework intermediate on the folding pathway of ribonuclease A[END_REF], the diffusioncollision model [START_REF] Karplus | Protein folding dynamics: the diffusion-collision model and experimental data[END_REF], the hydrophobic collapse model [START_REF] Rackovsky | Hydrophobicity, hydrophilicity, and the radial and orientational distributions of residues in native proteins[END_REF] or the nucleation and growth mechanism [START_REF] Fersht | Nucleation mechanism in protein folding[END_REF]. George Rose proposed a simple hierarchical model [START_REF] Rose | Hierarchic organization of domains in globular proteins[END_REF], which assemblies small units in a hierarchical manner [START_REF] Lesk | Folding units in globular proteins[END_REF]Baldwin and Rose 1999a;[START_REF] Matthews | Comparison of the predicted and observed secondary structure of T4 phage lysozyme[END_REF]Haspel et al. 2003a;[START_REF] Matthews | Comparison of the predicted and observed secondary structure of T4 phage lysozyme[END_REF] coupled with the hydrophobic effect as the driving force [START_REF] Dill | Theory for the folding and stability of globular proteins[END_REF][START_REF] Dill | From Levinthal to pathways to funnels[END_REF].

It leads to the construction of protein domains and complete folds.

Analyzing protein structures in terms of protein domains has been a long and fruitful research area for a long time. Many different approaches have been proposed [START_REF] Holm | Parser for protein folding units[END_REF][START_REF] Siddiqui | Continuous and discontinuous domains: an algorithm for the automatic generation of reliable protein domain definitions[END_REF][START_REF] Swindells | A procedure for detecting structural domains in proteins[END_REF][START_REF] Holm | Dictionary of recurrent domains in protein structures[END_REF][START_REF] Wernisch | Identification of structural domains in proteins by a graph heuristic[END_REF][START_REF] Xu | Protein domain decomposition using a graphtheoretic approach[END_REF][START_REF] Anselmi | Identification of protein domains on topological basis[END_REF][START_REF] Dengler | Protein structural domains: analysis of the 3Dee domains database[END_REF][START_REF] Siddiqui | 3Dee: a database of protein structural domains[END_REF][START_REF] Alexandrov | PDP: protein domain parser[END_REF][START_REF] Guo | Improving the performance of DomainParser for structural domain partition using neural network[END_REF][START_REF] Emmert-Streib | A topological algorithm for identification of structural domains of proteins[END_REF][START_REF] Joshi | A Decade of Computing to Traverse the Labyrinth of Protein Domains[END_REF][START_REF] Zhou | DDOMAIN: Dividing structures into domains using a normalized domain-domain interaction profile[END_REF]. They are based on numerous processes and algorithms. Most of them had initially an available website, but surprisingly at the time of this review none is functional.

Protein domains are also evolutionary units of proteins. The prediction of protein domains from sequence information can improve tertiary structure prediction [START_REF] Chivian | Automated prediction of CASP-5 structures using the Robetta server[END_REF]) and enhance protein function annotation [START_REF] Holland | Partitioning protein structures into domains: why is it so difficult?[END_REF]), but domains also been used to help structure determination [START_REF] Campbell | Building protein structure and function from modular units[END_REF] In the same way, our database of pre-cutting proteins provides useful materials for further analysis on structure, size, composition in amino acid and secondary structures of protein units. Such experiments open the way to other ambitious development like construction of three dimensional structures of proteins with protein units as it has been shown with similar approaches (Haspel et al. 2003a;[START_REF] Inbar | Protein structure prediction via combinatorial assembly of sub-structural units[END_REF]. As shown, Protein Units a valuable tools to understand protein folding, predict protein structure, identify structural domains. Futures developments will concern mainly the use of PUs for classification and for prediction purposes.

  Different languages and softwares are used. The web page in HTML shows on the upper www.dsimb.inserm.fr/dsimb_tools/peeling/). After the submission of PDB file by the user, the common gateway interface (CGI) gets the values from the web form and transmitting it to the perl core instance. Then PDB file undergoes appropriate treatment. It start by the cleaning of the PDB files to ensure a correct format and afterward by the launching of secondary structure assignment done by DSSP software(Kabsch and Sander

  are dedicated to visualize (i) the hierarchical peeling of the protein structures, (ii) the probabilities contact matrix and (iii) schematic representation of PUs in sequence with their contents in secondary structures. Two visualization softwares are used:(i) Ray tracing proteins structures relies on PyMol (DeLano 2002) which gives excellent rendering. The perl core creates a dedicated PyMol script which is used and is also given to the user which can adapt to its own needs. The format conversion and the post-rendering of the pictures was managed by ImageMagick suite (ImageMagick).

Figures 4

 4 Figures 4 and 5 show the cutting of dialkylglycine decarboxylase through Protein

  quality. In the same way, the different Figures generated through R software, (i) dendogram showing the entire process of splitting, (ii) the presentation of PUs with secondary structures and (iii) contact map with delineation of PUs, are an efficient representation. All these points make the Protein Peeling web server a unique tool to analyze protein structures.

FiguresFigure 1 .

 1 Figures

  Figure 1. DIAL web server. (a) the entrance page of DIAL where the use upload the PDB file,

Figure 2 .

 2 Figure 2. Principle of PP2 web server. All the different steps of the process are presented

Figure 3 .

 3 Figure 3. Web page of Protein Peeling 2. On the left part is given the different information

Figure 4 .

 4 Figure 4. PP2 cutting process of dialkylglycine decarboxylase. (a) Representation of the protein sequence with delineation of the different PUs in different colors. Are also given the secondary structure assignment done by DSSP (Kabsch and Sander 1983). (b) Dendograms of the PP2 cutting. Is shown for each R 2 value the number of generated PUs (Ihaka and Gentleman 1996).

Figure 5 .

 5 Figure 5. PP2 cutting process of dialkylglycine decarboxylase. Are shown all the different PUs generated for a very high R 2 value.

  , guide protein engineering[START_REF] Guerois | Protein design based on folding models[END_REF] and mutagenesis[START_REF] Nielsen | Identification of cell-binding sites on the Laminin alpha 5 N-terminal domain by site-directed mutagenesis[END_REF]. mutagenesis analysis and protein engineering. Other are more specialized as OPUS-Dom[START_REF] Wu | OPUS-Dom: applying the foldingbased method VECFOLD to determine protein domain boundaries[END_REF]), a de novo method for predicting protein domain boundaries. Its methodology is based on a coarse-grained folding method, which constructs low-resolution structural models from a target sequence by folding a chain of vectors representing the predicted secondary-structure elements.Analyzing protein structures in terms of small compact protein units is a less common research. As we shown only two methods are available at this time, DIAL and Protein Peeling. Interest of DIAL is the proposition of potential alternative splitting events. Interest of Protein Peeling is the availability of numerous options allowing a real expertise of the protein structure. Moreover, visualization tools allow a direct analysis of the cutting through JMol applet while PyMol script and accompanying Figures are of great

	Hence, some approaches have been proposed to predict protein structural domains from the
	sole	knowledge	of	the	sequence.	For	instance,	DOMAC
	(http://www.bioinfotool.org/domac.html) is a hybrid domain prediction web service

integrating template-based and ab initio methods

[START_REF] Cheng | DOMAC: an accurate, hybrid protein domain prediction server[END_REF]

. Its template-based method is accurate enough for guiding protein structure prediction, structure determination, function annotation,

Acknowledgments

This work was supported by grants from the Ministère de la Recherche, Université Paris Diderot -Paris 7, National Institute for Blood Transfusion (INTS) and the National Institute for Health and Medical Research (INSERM).


Seasonal flow variability of a temperate glacier in the Mont Blanc massif observed by GPS

Fanny Ponton (1), Andrea Walpersdorf (1), Michel Gay (2), Emmanuel Trouvé (3), Jean-Louis Mugnier (1), Nathalie Cotte (1), Laurent Ott (2) and Jonathan Serafini [START_REF] Tregoning | Accuracy of absolute precipitable water vapor estimates from GPS observations[END_REF][START_REF] Herring | Introduction to GAMIT/GLOBK Release 10[END_REF] (1) ISTerre, CNRS, Université Joseph Fourier, Maison des Géosciences, BP 53, 38 041 Grenoble Cedex 9, France (fanny.ponton@ujf-grenoble.fr; andrea.walpersdorf@ujf-grenoble.fr; Jean-Louis.Mugnier@univ-savoie.fr; Nathalie.Cotte@ujfgrenoble.fr), [START_REF] Walpersdorf | Assessment of GPS data for meteorological applications over Africa: Study of error sources and analysis of positioning accuracy[END_REF] GIPSA-lab, INPG, Université Joseph Fourier, 961 rue de la Houille Blanche, 38402 Saint Martin d'Hères, France (michel.gay@gipsa-lab.grenoble-inp.fr; laurent.ott@gipsa-lab.grenoble-inp.fr), (3) LISTIC, Polytech Annecy-Chambéry, BP 80439, 74944 Annecy-le-Vieux cedex, France (emmanuel.trouve@univ-savoie.fr), [START_REF] Herring | Introduction to GAMIT/GLOBK Release 10[END_REF] now at GePaSud, Université de Polynésie française, B.P. 6570, 98702 FAA'A, Tahiti -Polynésie Française (jonathan.serafini@upf.pf).

ABSTRACT

We present four years of GPS data acquired continuously on the Argentière glacier in the Mont Blanc massif, France. Our local permanent GPS network is composed of two stations on the glacier at altitudes of 2441 m and 2770 m, and two stations in static places, one in the valley of Chamonix at an altitude of 1121 m and the other on a rock outcrop near the glacier at the altitude of 2835 m. The measurements yield average displacement rates of about 13 cm/day for the upper glacier station (2770 m) and 17 cm/day for the lower site (2441 m). Moreover, we observe an annual variability of the displacement rates of up to 28 %, with fastest flow in late summer. The continuous monitoring of the Argentière glacier flow over several annual cycles can now be used to determine the correlation with climatological parameters such as temperature and cumulated precipitations.

INTRODUCTION

The study of the Argentière glacier in the Mont Blanc massif in France by continuous GNSS measurements is part of the EFIDIR project (Extraction and Fusion of Information for ground displacement measurements with Radar Imagery). EFIDIR is funded by the French ANR from 2008 to 2011 to develop methodologies for improving the precision of radar imagery for displacement measurements. The project brings together researchers in different domains as signal processing, space geodesy and geophysics. The Argentière glacier has been selected as one of the EFIDIR targets representative for a fast moving geophysical object. Satellite radar images have been collected from TerraSAR-X and Radarsat2 satellites since the end of 2007 (for coverage see Fig. 1). Several corner reflectors have been set up to create well defined points in the radar images. Continuous GPS measurements in the same region, in particular on the Argentière glacier will be used for two different aims. First, the GPS stations on the glacier provide the ground truth for displacement measurements by radar imagery. One of the stations located close to a corner reflector helps establishing the spatial correlation between successive images. Second, the GPS network will be used to provide in situ measurements of the tropospheric delay that affects both the GPS and the radar imagery in the same way. While tropospheric delay remains one of the major error sources in satellite radar imagery, the analysis of GPS data permits to quantify this highly variable delay precisely (e.g. [START_REF] Tregoning | Accuracy of absolute precipitable water vapor estimates from GPS observations[END_REF][START_REF] Walpersdorf | Assessment of GPS data for meteorological applications over Africa: Study of error sources and analysis of positioning accuracy[END_REF]). The development of an optimal use of this precise but localized information provided by the GPS network to correct radar images is one of the goals of the EFIDIR project. Sub-products of the GNSS instrumentation on the Argentière glacier are the times series of positions of two stations installed on the ice sheet. The relative continuous time series will help to identify not only an average flow velocity but also a seasonal flow variability. This is new information for temperate glaciers where altitude, climatic conditions and difficulties to access limit the possibilities to obtain continuous and numerous datasets. These new observations on the Argentière glacier flow could be linked with climatic conditions and temperate glacier flow models. The glacier of Argentière in the Mont Blanc massif has a length of about 10 km from the edge of the tongue at the altitude of 1600 m to the top part at the altitude of 3400 m (Fig. 2). In 2003, it covered a surface of 12.4 km². In the framework of the EFIDIR project, we developed a local GPS network to monitor this glacier and the overlying troposphere. The network is composed of 4 stations installed at different altitudes. To provide ground truth for the surface ice flow velocities to be measured by radar satellites, two of them are installed on the ice, at 2770 m and 2441 m of altitude (ARGG and SERA). The upper one is close to the line of balance between snow accumulation and melt of the glacier, where also one of the corner reflectors is installed. Two more stations are located outside of the glacier, one on rock outcrops near the Argentière mountain hut at 2835 m (ARGR), and one in the close by town of Chamonix at 1121 m (CHMX) (Fig. 1 and2). In this work, we present the GPS data analysis leading to a consistent position time series for the upper glacier station, ARGG. The analysis of the second glacier station, SERA, is still under work. Only the very first data from October 2008 have already been analyzed in the framework of the MSc thesis of J. Serafini at ISTerre [START_REF] Serafini | Analyse des données GPS du glacier d'Argentière[END_REF]. We cite these results here in the meanwhile for completeness (last line in Tab.1).

Figure 1. Map of the local GPS network. The color code indicates the original networks of the stations (RENAG, RGP, INGV (RING), AGNES). The frames show the surface covered by the Terra-SAR X and

GPS DATA AND DATA ANALYSIS

The local GPS network provides data since spring 2007 (ARGG since 06/2007, ARGR and CHMX since 07/2007 and SERA since 11/2008). The first glacier station ARGG was initially installed on the edge of the corner reflector. Re-orientations of the corner reflector for specific satellite overflows caused several discontinuities in the coordinate time series. Although the corner reflector was anchored in the ice, differential ice melting in summer probably tilted the corner reflector and created motions of the GPS antenna that cannot be attributed to glacier flow. To obtain more consistent position time series, the antenna was finally mounted on a wooden mast at a few meters distance from the corner reflector. This mast is about 6 m long with 2.5 m fixed in the ice at the time of installation (Fig. 3). Ice melt in summer of 1 -2 m necessitates regular re-installations of the mast. These different manipulations of the antenna setup led to discontinuities in the coordinate time series that have to be taken into account when establishing average flow velocities. The SERA station was mounted on a mast from the beginning. Due to the larger amount of ice melt in the lower part of the glacier, this station setup had to be manipulated even more often than for the ARGG station. The GPS equipment used are Topcon GB1000 receivers with PG-A1 antennae for ARGG and SERA, Trimble NetRS with a Zephyr antenna for ARGR and Ashtech ZXII with a Dorne Margolin choke-ring antenna for CHMX. The GPS data analysis was done with MIT's GAMIT/GLOBK software version 10.4 [START_REF] Herring | Introduction to GAMIT/GLOBK Release 10[END_REF]. Average positions are evaluated over 6 hour sessions. This duration seems the most appropriate for the large glacier displacements, without weakening too much the positioning precision for the stable sites. The network analyzed is composed of the 4 local GPS stations, 34 stations from the regional or national networks RENAG (http://webrenag.unice.fr), RGP (http://rgp.ign.fr), AGNES (http://www.swisstopo.admin.ch) and RING (http://ring.gm.ingv.it), and 13 stations from the international IGS network (http://igscb.jpl.nasa.gov). In the analysis of the 6 h sessions, the troposphere is modeled using the tropospheric mapping function VMF1 [START_REF] Boehm | Troposphere mapping functions for GPS and very long baseline interferometry from European Centre for Medium Range Weather Forecasts operational analysis data[END_REF] and its pressure and temperature values as a priori values for each station, to estimate one zenith delay parameter every 2 hours and one horizontal tropospheric gradient per 6 h session. Atmospheric and ocean loading are modeled according to [START_REF] Tregoning | Effects of atmospheric pressure loading and seven parameter transformations on estimates of geocenter motion and station heights from space geodetic observations[END_REF] and by the FES2004 model [START_REF] Lyard | Modelling the global ocean tides: modern insights from FES2004[END_REF]. Resulting average position time series are represented in the ITRF 2008 reference frame [START_REF] Altamimi | ITRF2008: an improved solution of the international terrestrial reference frame[END_REF], thanks to the IGS stations included in the analysis. The resulting time series for the glacier stations contain several gaps and discontinuities, due to power failures and the above mentioned antenna setup manipulations. We estimated and corrected for the displacements due to manipulations by extrapolating the antenna motion from the last position using the average flow rate of the data segments before and after any discontinuity (jump or gap), and evaluating the "manual" displacement by the difference of this extrapolated position and the first position of the succeeding segment (see schematic representation in Fig. 4). For applying this strategy, 15 continuous data segments have been identified and the average flow velocities have been determined (Tab. 1). The extrapolation between successive segments showed that only 4 manipulations could be clearly identified and related to interventions on site (e.g. installation and re-installations of the mast). The 4 offsets (usually of several meters) have been corrected for in the position time series shown hereafter (Fig. 5 and6).

POSITION TIME SERIES

The position time series of the ARGG station are shown on Fig. 5. An average displacement rate has been evaluated by component, with 6.0 cm/day toward the north, 11.9 cm/day toward the west and a downward motion of 1.1 cm/day. The flow direction is well aligned with the direction of the glacial valley (Fig. 2). With respect to these large linear displacements, any residual motion is hardly visible. To highlight any additional accelerations and decelerations, the average displacement rate has been subtracted from the three position components, with the resulting de-trended time series shown in Fig. 6. Seasonal variations of the glacier flow are now clearly visible. In particular, a regular annual signal seems to be dominating since 2008, while the 2007 data are less coherent, maybe due to the antenna mount on the corner reflector. The largest flow velocities are found rather simultaneously on the three components, in north, west and downward direction, around the months of August and September each year. The amplitudes of the position variations are about 80 cm on the North component, about 140 cm on the East component and 40 cm on the vertical. These differences in amplitude correspond to the direction of motion with a large west component, a smaller north component and a relatively tiny downward motion. Eventually, in addition to the annual variation, another pluri-annual signal is superposed in the time series that has to be confirmed by a longer measurement span.

DISCUSSION

One methodological test highlighting the coherence of the GPS measurements and their significance for monitoring the glacier flow is the comparison of the horizontal flow velocity with the simultaneous vertical motion. Indeed, if the experimental setup is well adapted (and the motion of the GPS antenna representative for the glacier motion), the relation between horizontal and vertical displacements should correspond to the slope of the glacier at the place of the GPS station. The evaluation of the slope of the motion has been done for each continuous data segment (Tab. 1). The limitation is the low vertical displacement rate, with the vertical position being less well constraint by the GPS measurements, due to the large elevation masks by the mountain ranges around the GPS station. However, the average vertical displacement rate is still determined to better than 1 mm/day for each of the segments. For comparison, the uncertainty of the horizontal average velocities is 0.3 mm/day. The slopes evaluated for each segment vary from 13 to 2 degrees, while the average velocities over the whole observation span yield a slope of 5 degrees. Indeed, high slopes are observed in the beginning of the experiment, in summer 2007, when the GPS antenna was still mounted on the upper edge of the corner reflector. A tilt of the corner reflector could explain the too large downward motions at this epoch. Since 2008, and in particular since the antenna setup on the mast in July 2008, the slopes for each data segment yield very coherent values of 4-5 degrees, while the total velocities vary by 4.2 cm/day. This shows that the measurement of the velocity variations is significant.

The average flow velocity at the ARGG station at 2770 m altitude should be compatible with annual measurements of several profiles across the Argentière glacier presented by [START_REF] Vincent | Glacier thickening and decay analysis from 50 years of glaciological observations performed on Glacier d'Argentière, Mont Blanc area, France[END_REF]. One profile situated at 2730 m is close to ARGG station so that velocities can be compared. According to [START_REF] Vincent | Glacier thickening and decay analysis from 50 years of glaciological observations performed on Glacier d'Argentière, Mont Blanc area, France[END_REF], the flow velocity at this altitude has been rather stable over the last 25 years, around 55 m/yr with variations of ± 5 m/yr correlated over several years. In particular, the annual rate decreased between 2000 and 2007 from 58 to 50 m/yr, corresponding to 15.9 to 13.7 cm/day. Our measurement of an average velocity from 2007 to 2010 of 13.3 cm/day is coherent with these former results. As the GPS station is at an altitude 40 m above the profile, the GPS flow velocity should be slightly lower than the velocity at the profile. This could indicate that the flow velocity probably did not decrease since the last date published in [START_REF] Vincent | Glacier thickening and decay analysis from 50 years of glaciological observations performed on Glacier d'Argentière, Mont Blanc area, France[END_REF]. The indication of an eventual long term variation that starts to be visible in the GPS time series is also in agreement with the pluri-annual variability shown by [START_REF] Vincent | Glacier thickening and decay analysis from 50 years of glaciological observations performed on Glacier d'Argentière, Mont Blanc area, France[END_REF]. Our continuous GPS measurements resolve for the first time the short term variability of the Argentière glacier flow. As shown above, we can provide evidence for an annual signal, with a maximum velocity at the end of summer. The departure from mean velocity reaches 28 % in our observation interval (Tab. 1). The comparison with the variability of ground temperature in the study zone (Fig. 6 bottom) shows that this maximum is reached at the end of the period of maximum temperature in summer (mid-September). Unfortunately, until now, only in 2008 this summer period is continuously measured, but the change in flow velocity seems to take place instantaneously after the drop of ground temperature. The amplitude of the seasonal variations as well as the particular signature of the variations are new observations that could be exploited for improving glacier flow models.

  Figure 1. Map of the local GPS network. The color code indicates the original networks of the stations (RENAG, RGP, INGV (RING), AGNES). The frames show the surface covered by the Terra-SAR X and Radarsat 2 images.

Figure 2 .

 2 Figure 2. Local view of the Argentière glacier with locations of dedicated GPS stations. The average flow velocities of the two glacier stations ARGG and SERA are indicated schematically, evaluated to 13.3 cm/day (in this work) and 16.8 cm/day [3].

Figure 3 .

 3 Figure 3. Photo of the ARGG GPS station.

Figure 4 .

 4 Figure 4. Schematic representation of the strategy used to correct for offsets in the position time series due to manipulations of the antenna setup.

Figure 5 .

 5 Figure 5. Position time series of station ARGG with respect to ITRF 2008. The three boxes show the North, East and Up component, respectively. Average flow rates are indicated for each component. The dashed vertical line indicates the moment when the antenna was un-mounted from the corner reflector and installed on the measurement mast.

Figure 6 .

 6 Figure 6. Detrended position time series of the ARGG station. The dashed blue vertical line highlights the installation of the mast. The red vertical lines indicate

Table 1 .

 1 Velocities per data segment

		N	E	U	total	slope epoch	# obs.
	ARGG [cm/day]			[ ]	[yr]	[days]
	total 6.0 -11.9 -1.1	13.3	5	2007.49 446
	seg. 1	6.3 -12.1 -2.5	13.9	10	2007.49 33
	2	3.6 -11.6 -2.8	12.4	13	2007.58 22
	3	5.3 -15.7 -3.6	17.0	12	2007.65 10
					(+28%)
	4	3.9 -11.4 -1.4	12.1	7	2007.68 35
	5	5.4 -10.3 -0.4	11.7	2	2008.16 16
					(-12%)
	6	6.0 -11.3 -0.8	12.8	4	2008.20 16
	7	6.9 -13.5 -1.0	15.2	4	2008.36 10
	8	6.2 -12.3 -1.2	13.8	5	2008.56 5
	9	5.9 -13.1 -1.3	14.4	5	2008.58 57
	10	5.5 -10.7 -0.9	12.1	4	2008.74 20
	11	5.6 -10.7 -0.9	12.1	4	2008.81 35
	12	7.5 -14.4 -1.4	16.3	5	2009.41 25
	13	6.4 -12.4 -1.2	14.0	5	2009.47 38
	14	5.8 -11.2 -0.9	12.7	4	2009.95 47
	15	6.9 -13.4 -1.1	15.1	4	2010.26 79
	SERA 9.5 -13.8 -0.3	16.8	1	2008.81 14


Thierry Huillet

Martin Möhle

ERRATUM ON 'POPULATION GENETICS MODELS WITH SKEWED FERTILITIES: A FORWARD AND BACKWARD ANALYSIS'

Keywords: Bell polynomials, compound Poisson model, Dirichlet model, Kingman coalescent, saddle point method, Wright-Fisher model 2010 Mathematics Subject Classification: Primary 60J10, Secondary 60K35, 92D10, 92D25

The article 'Population genetics models with skewed fertilities: a forward and backward analysis ', Stochastic Models 27, 521-554 (2011) contains on top of page 536 a formula for the joint factorial moments of the offspring numbers µ1, . . . , µN , which is wrong in that generality. It is clarified for which compound Poisson models this formula holds true. It turns out that the only compound Poisson models for which this formula holds true are skewed generalized Wright-Fisher models and skewed generalized Dirichlet models. An erratum is provided correcting the results in Section 4 of the mentioned article from Proposition 4.2 on. The main conclusion (Theorem 3.2) that many symmetric compound Poisson population models are in the domain of attraction of the Kingman coalescent, remains valid, however, its proof turns out to be more involved.

A key analytic tool in the proof is the saddle point method. In particular, the correct time-scaling (effective population size) is provided.

Introduction

Let us briefly recall the definitions of conditional branching process models and compound Poisson models. Conditional branching process models are population models with fixed population size N ∈ N := {1, 2, . . .} and non-overlapping generations. They are defined in terms of a sequence (ξ n ) n∈N of independent non-negative integer-valued random variables satisfying P(ξ 1 + • • • + ξ N = N ) > 0. If, for i ∈ {1, . . . , N }, µ N,i denotes the number of offspring of the ith individual alive in some fixed generation, then the random variables µ N,1 , . . . , µ N,N have (by definition) joint distribution

We now turn to the definition of compound Poisson population models. Let θ 1 , θ 2 , . . . be strictly positive real numbers and let φ(z) = ∞ m=1 φ m z m /m!, |z| < r, be a power series with radius r ∈ (0, ∞] of convergence and with nonnegative coefficients φ m ≥ 0, m ∈ N. It is also assumed that φ 1 > 0. Compound Poisson models are particular conditional branching process models where each random variable ξ n has probability generating function (pgf)

f n (x) := E(x ξn ) = exp(-θ n (φ(z) -φ(zx))), |x| ≤ 1. (1)
In [START_REF] Berestycki | Gibbs distributions for random partitions generated by a fragmentation process[END_REF], z is viewed as a fixed parameter satisfying |z| < r. However, in order to analyze compound Poisson models it is useful to view z as a variable and to introduce, for θ ∈ [0, ∞), the Taylor expansion

exp(θφ(z)) = ∞ k=0 σ k (θ) k! z k , |z| < r.
The coefficients σ k (θ) depend on (φ m ) m∈N and they satisfy the recursion σ 0 (θ) = 1 and

σ k+1 (θ) = θ k l=0 k l φ k-l+1 σ l (θ), k ∈ N 0 , θ ∈ [0, ∞). (2)
The coefficients σ k (θ) are mainly introduced, since, by [START_REF] Berestycki | Gibbs distributions for random partitions generated by a fragmentation process[END_REF], the distribution of ξ n , n ∈ N, satisfies

P(ξ n = k) = σ k (θ n ) z k k! exp(-θ n φ(z)), k ∈ N 0 . (3)
From φ 1 > 0 it follows that σ k (θ) is a polynomial in θ of degree k. In the literature (see, for example, [START_REF] Berestycki | Gibbs distributions for random partitions generated by a fragmentation process[END_REF] or [START_REF] Charalambides | An asymptotic formula for the exponential polynomials and a central limit theorem for their coefficients[END_REF]) the σ k (θ) are called the exponential polynomials. We have σ 1 (θ) = θφ 1 , σ 2 (θ) = θφ 2 +θ 2 φ 2 1 , σ 3 (θ) = θφ 3 +3θ 2 φ 1 φ 2 +θ 3 φ 3 1 , σ 4 (θ) = θφ 4 +θ 2 (4φ 1 φ 3 +3φ 2 2 )+6θ 3 φ 2 1 φ 2 + θ 4 φ 4 1 , and so on. The coefficients B kl (φ 1 , φ 2 , . . .), k ∈ N 0 , l ∈ {0, . . . , k}, of the polynomials σ k (θ) = k l=0 B kl (φ 1 , φ 2 , . . .) θ l , k ∈ N 0 , are called the Bell coefficients. It is readily checked that ξ n has descending factorial moments

E((ξ n ) k ) = f (k) n (1) = z k k l=0 B kl (φ (z), φ (z), . . .) θ l n , n ∈ N, k ∈ N 0 , i.e. E(ξ n ) = θ n zφ (z), E((ξ n ) 2 ) = θ n z 2 φ (z)+θ 2 n z 2 (φ (z))
2 and so on. The descending factorial moments therefore satisfy the recursion

E((ξ n ) k+1 ) = θ n k l=0 k l z k-l+1 φ (k-l+1) (z)E((ξ n ) l ), n ∈ N, k ∈ N 0 .
Throughout the article, for x ∈ R and k ∈ N 0 , the notations (x

) k := x(x -1) • • • (x -k + 1) and [x] k := x(x + 1) • • • (x + k -1)
are used for the descending and ascending factorials respectively, with the convention that (x) 0 := [x] 0 := 1.

Results

We derive the correct formula for the joint factorial moments of the offspring random variables µ 1 , . . . , µ N for an arbitrary compound Poisson population model. It is known (see, for example, [7, p. 535]) that µ = (µ 1 , . . . , µ N ) has distribution

P(µ = j) = N ! σ N (Θ N ) N n=1 σ jn (θ n ) j n ! , j = (j 1 , . . . , j N ) ∈ ∆(N ), (4)
where

Θ N := θ 1 + • • • + θ N and ∆(N ) denotes the discrete N -simplex consisting of all j = (j 1 , . . . , j N ) ∈ N N 0 satisfying j 1 + • • • + j N = N .
Note that the distribution of µ is not necessarily exchangeable. It follows that µ has joint factorial moments

E((µ 1 ) k1 • • • (µ N ) k N ) = j=(j1,...,j N )∈∆(N ) (j 1 ) k1 • • • (j N ) k N P(µ = j) = N ! σ N (Θ N ) j 1 ,...,j N ∈N 0 j 1 +•••+j N =N (j 1 ) k1 • • • (j N ) k N σ j1 (θ 1 ) • • • σ j N (θ N ) j 1 ! • • • j N ! = N ! σ N (Θ N ) j 1 ≥k 1 ,...,j N ≥k N j 1 +•••+j N =N σ j1 (θ 1 ) • • • σ j N (θ N ) (j 1 -k 1 )! • • • (j N -k N )! , (5)
N ∈ N, k 1 , . . . , k N ∈ N 0 , θ 1 , . . . , θ N ∈ (0, ∞)
, which is not a simple expression. However, it is known that for some compound Poisson models, for example for the Wright-Fisher model, the alternative and simpler formula

E((µ 1 ) k1 • • • (µ N ) k N ) = (N ) k σ k1 (θ 1 ) • • • σ k N (θ N ) σ k (θ 1 + • • • + θ N ) (6)
holds for all k 1 , . . . , k N ∈ N 0 and all θ 1 , . . . , θ N ∈ (0, ∞), where k

:= k 1 + • • • + k N .
In the following it is clarified which compound Poisson models satisfy [START_REF] Graham | Concrete Mathematics. A Foundation for Computer Science, Second Edition[END_REF]. Our results are based on the following basic but fundamental lemma, which essentially coincides with [9, Lemma 2.1], however, we prefer here to state it slightly different.

Lemma 2.1 A compound Poisson model (with given fixed power series φ) satisfies the factorial moment formula [START_REF] Graham | Concrete Mathematics. A Foundation for Computer Science, Second Edition[END_REF] for all population sizes N ∈ N, all k 1 , . . . , k N ∈ N 0 and all θ 1 , . . . , θ N ∈ (0, ∞) if and only if the coefficients σ k (θ), k ∈ N 0 , θ ∈ (0, ∞), satisfy the relation

σ k+1 (θ) σ k (θ) + σ k +1 (θ ) σ k (θ ) = σ k+k +1 (θ + θ ) σ k+k (θ + θ ) , k, k ∈ N 0 , θ, θ ∈ (0, ∞). (7)
Proof. Suppose first that (7) holds. We essentially follow the proof of [START_REF] Möhle | Coalescent processes derived from some compound Poisson population models[END_REF]Lemma 2.1]. By induction on N ∈ N it follows that

N j=1 σ kj +1 (θ j ) σ kj (θ j ) = σ k+1 (θ 1 + • • • + θ N ) σ k (θ 1 + • • • + θ N ) , k 1 , . . . , k N ∈ N 0 , θ 1 , . . . , θ N ∈ (0, ∞). (8)
Let us now verify (6) by backward induction on k = k 1 + • • • + k N . For k > N both sides of (6) are equal to zero. For k = N we have

E((µ 1 ) k1 • • • (µ N ) k N ) = k 1 ! • • • k N !P(µ 1 = k 1 , . . . , µ N = k N ) = N ! σ k1 (θ 1 ) • • • σ k N (θ N ) σ k (θ 1 + • • • + θ N ) ,
which is [START_REF] Graham | Concrete Mathematics. A Foundation for Computer Science, Second Edition[END_REF] for k = N . The backward induction step from k + 1 to k (∈ {0, . . . , N -1}) works as follows. From µ 1 + • • • + µ N = N and by induction it follows that

(N -k)E((µ 1 ) k1 • • • (µ N ) k N ) = E((µ 1 ) k1 • • • (µ N ) k N N j=1 (µ j -k j )) = N j=1 E((µ 1 ) k1 • • • (µ j ) kj +1 • • • (µ N ) k N ) = N j=1 (N ) k+1 σ k1 (θ 1 ) • • • σ kj +1 (θ j ) • • • σ k N (θ N ) σ k+1 (θ 1 + • • • + θ N ) = (N ) k+1 σ k1 (θ 1 ) • • • σ k N (θ N ) σ k+1 (θ 1 + • • • + θ N ) N j=1 σ kj +1 (θ j ) σ kj (θ j ) = (N ) k+1 σ k1 (θ 1 ) • • • σ k N (θ N ) σ k (θ 1 + • • • + θ N ) , (9)
where the last equality holds by [START_REF] Möhle | Total variation distances and rates of convergence for ancestral coalescent processes in exchangeable population models[END_REF]. Division of (9) by N -k shows that (6) holds for k which completes the backward induction. Conversely, if [START_REF] Graham | Concrete Mathematics. A Foundation for Computer Science, Second Edition[END_REF] holds for all N ∈ N, k 1 , . . . , k N ∈ N 0 and all θ 1 , . . . , θ N ∈ (0, ∞), then

(N ) k+1 σ k1 (θ 1 ) • • • σ k N (θ N ) σ k (θ 1 + • • • + θ N ) = (N -k)E((µ 1 ) k1 • • • (µ N ) k N ) = (N ) k+1 σ k1 (θ 1 ) • • • σ k N (θ N ) σ k+1 (θ 1 + • • • + θ N ) N j=1 σ kj +1 (θ j ) σ kj (θ j ) ,
where the last equation is obtained by doing the same calculations as above. Thus, (8) holds.

Choosing N = 2 in (8), it follows that [START_REF] Huillet | Population genetics models with skewed fertilities: a forward and backward analysis[END_REF] holds which completes the proof. 2 Relation (7) puts strong constrains on the coefficients σ k (θ), k ∈ N 0 , θ ∈ (0, ∞), and, consequently, on the power series φ of the compound Poisson model. Lemma 2.1 of [START_REF] Möhle | Coalescent processes derived from some compound Poisson population models[END_REF] and the remarks thereafter show that (7) holds if and only if φ(z) = φ 1 z or

φ(z) = - φ 2 1 φ 2 log 1 - φ 2 φ 1 z , |z| < φ 1 φ 2 . ( 10
)
The only compound Poisson models satisfying the factorial moment formula ( 6) for all N ∈ N, all k 1 , . . . , k N ∈ N 0 , and all θ 1 , . . . , θ N ∈ (0, ∞) are therefore generalized Wright-Fisher models (with φ of the form φ(z) = φ 1 z for some constant φ 1 ∈ (0, ∞)) and generalized Dirichlet models with φ of the form [START_REF] Möhle | A classification of coalescent processes for haploid exchangeable population models[END_REF] for some constants φ 1 , φ 2 ∈ (0, ∞). For all other compound Poisson models the formula (6) does not hold for all N ∈ N, all k 1 , . . . , k N ∈ N 0 and all θ 1 , . . . , θ N ∈ (0, ∞), a typical counter example being the compound Poisson model with power series φ of the form φ(z) = e z -1, or, equivalently, φ m = 1 for all m ∈ N.

Corrections and further results

We comment which parts of [START_REF] Huillet | Population genetics models with skewed fertilities: a forward and backward analysis[END_REF] need correction. The results in [7, Sections 5 and 6] are correct, since the models studied there satisfy [START_REF] Graham | Concrete Mathematics. A Foundation for Computer Science, Second Edition[END_REF]. The statements in [7, Section 7] 13)], then the backward process X of the associated skewed conditional branching process model has transition probabilities

P i,j = 1 N i 1≤n1<•••<nj ≤N k 1 ,...,k j ∈N 0 k:=k 1 +•••+k j ≤N (N ) k k 1 ! • • • k j ! • • ( j i=1 σ ki (θ ni )) σ N -k (Θ N - j i=1 θ ni ) σ N (Θ N ) l 1 ,...,l j ∈N l 1 +•••+l j =i k 1 l 1 • • • k j l j , i, j ∈ S, (11)
with the convention that P i,0 = δ i0 , i ∈ S. Here Θ N := θ 1 + • • • + θ N , S := {0, . . . , N }, and the coefficients σ k (θ), k ∈ N 0 , θ ∈ [0, ∞), are recursively defined via (2). In particular,

P i,1 = N n=1 N k=i N -i k -i σ k (θ n ) σ N -k (Θ N -θ n ) σ N (Θ N ) , i ∈ {1, . . . , N }. ( 12
)
Proof. For j ∈ {1, . . . , N }, pairwise distinct n 1 , . . . , n j ∈ {1, . . . , N }, and k 1 , . . . , k j ∈ N 0 with k

:= k 1 + • • • + k j ≤ N we have P(µ n1 = k 1 , . . . , µ nj = k j ) = ( j i=1 P(ξ ni = k i )) P( m∈[N ]\{n1,...,nj } ξ m = N -k) P(ξ 1 + • • • + ξ N = N ) = (N ) k k 1 ! • • • k j ! σ k1 (θ n1 ) • • • σ kj (θ nj ) σ N -k (Θ N - j i=1 θ ni ) σ N (Θ N )
and, therefore, for l 1 , . . . , l j ∈ N 0 ,

E j i=1 µ ni l i = k1,...,kj j i=1 k i l i (N ) k k 1 ! • • • k j ! σ k1 (θ n1 ) • • • σ kj (θ nj )σ N -k (Θ N - j i=1 θ ni ) σ N (Θ N )
,

where the sum k1,...,kj extends over all k 1 , . . . , k j ∈ N 0 satisfying k

:= k 1 + • • • + k j ≤ N .
Thus (11) follows from [7, Eq. ( 4)]. For j = 1, (11) reduces to (12). Alternatively, (12) follows as well via

P i,1 = N n=1 E((µ n ) i ) (N ) i = N n=1 N k=i (k) i (N ) i P(µ n = k) = N n=1 N k=i (k) i (N ) i N k σ k (θ n ) σ N -k (Θ N -θ n ) σ N (Θ N ) = N n=1 N k=i N -i k -i σ k (θ n ) σ N -k (Θ N -θ n ) σ N (Θ N ) . 2
Remark. Proposition 3.1 in particular yields the coalescence probability

c N := P 2,1 = N n=1 N k=2 N -2 k -2 σ k (θ n ) σ N -k (Θ N -θ n ) σ N (Θ N ) , N ≥ 2 (13)
and the probability

d N := P 3,1 = N n=1 N k=3 N -3 k -3 σ k (θ n ) σ N -k (Θ N -θ n ) σ N (Θ N ) , N ≥ 3 (14)
that three individuals share a common parent. For the unbiased case, when all the parameters θ n = θ are equal to some constant θ ∈ (0, ∞), (13) and ( 14) reduce to

c N = N N k=2 N -2 k -2 σ k (θ) σ N -k ((N -1)θ) σ N (N θ) (15)
and

d N = N N k=3 N -3 k -3 σ k (θ) σ N -k ((N -1)θ) σ N (N θ) . (16)
It is clear that Theorem 4.3 of [START_REF] Huillet | Population genetics models with skewed fertilities: a forward and backward analysis[END_REF] does not hold for arbitrary symmetric compound Poisson models. Theorem 4.3 of [START_REF] Huillet | Population genetics models with skewed fertilities: a forward and backward analysis[END_REF] should be replaced by the following Theorem 3.2, which clarifies that many symmetric compound Poisson models are in the domain of attraction of the Kingman coalescent.

Theorem 3.2 Fix θ ∈ (0, ∞) and suppose that the equation θzφ (z) = 1 has a real solution z(θ) ∈ (0, r), where r ∈ (0, ∞] denotes the radius of convergence of φ. Then, µ N,1 → X in distribution as N → ∞, where X is a nonnegative integer valued random variable with distribution

P(X = k) = σ k (θ) (z(θ)) k k! e -θφ(z(θ)) , k ∈ N 0 . (17
:= 1/E((X) 2 ) = 1/(1 + θ(z(θ)) 2 φ (z(θ))) ∈ (0, 1].
Remarks. 1. If φ (r-) = ∞, then for all θ ∈ (0, ∞) the equation θzφ (z) = 1 has a real solution z(θ) ∈ (0, r). If φ (r-) < ∞, then a real solution z(θ) ∈ (0, r) of the equation θzφ (z) = 1 exists if and only if θrφ (r-) > 1. A concrete model satisfying φ (r-) < ∞ is provided in Example 4.8 in Section 4. The solution z(θ) ∈ (0, r) (if it exists) is unique since the map z → zφ (z) is strictly monotone increasing on (0, r). 2. The following proof even shows the convergence of all moments E(µ p N,1 ) → E(X p ) as N → ∞, p > 0. The distribution of the limiting variable X coincides with the distribution (3) of ξ 1 with the parameter z in (3) replaced by z(θ). Note that X has mean E(X) = θz(θ)φ (z(θ)) = 1. Thus, conditioning ξ 1 on the event that ξ 1 + • • • + ξ N = N and afterwards taking N → ∞, has altogether the effect that we 'nearly' recover the distribution of ξ 1 . We only loose the information about the mean of ξ 1 . 3. Closed expressions for the solution z = z(θ) of the equation u(z) := zφ (z) = 1/θ seem to be not available in general. By the inversion formula of Lagrange (see, for example, [4, Section 3.8]), for k ∈ N and x ∈ (0, ∞),

[x k ]u -1 (x) = [z k-1 ](z/u(z)) k k = [z k-1 ](1/φ (z)) k k = B k-1,k (ψ 0 , ψ 1 , . . .),
where ψ n := ψ (n) (0), n ∈ N 0 , with ψ(z) := 1/φ (z). Choosing x := 1/θ and noting that z(θ) = u -1 (x) yields the formal expansion

z(θ) = ∞ k=1 [z k-1 ](ψ(z)) k k θ -k = ∞ k=1 B k-1,k (ψ 0 , ψ 1 , . . .)θ -k .
Note however that, depending on φ and θ, this series does not necessarily need to converge, so we can only speak about a formal expansion here.

Proof. Fix k, l ∈ N 0 and θ ∈ (0, ∞). Let us verify that

σ n-k ((n -l)θ) (n -k)! ∼ (a(θ)) n √ 2πn b kl (θ), n → ∞, (18)
where a(θ) := e θφ(z(θ)) z(θ) and b kl (θ) := (z(θ)) k e -lθφ(z(θ))

1 + θ(z(θ)) 2 φ (z(θ)) . ( 19
)
We proceed similarly as in the proof of Theorem 2.1 of [START_REF] Charalambides | An asymptotic formula for the exponential polynomials and a central limit theorem for their coefficients[END_REF]. However, note that in [START_REF] Charalambides | An asymptotic formula for the exponential polynomials and a central limit theorem for their coefficients[END_REF], asymptotic expansions for σ n (θ) are provided whereas we are essentially interested in asymptotic expansions of σ n (nθ). By Cauchy's integral formula, σ n (θ)/n! = (2πi) -1 C z -(n+1) e θφ(z) dz, n ∈ N 0 , where C is some contour around the origin. Replacing n by n -k and θ by (n -l)θ it follows that

σ n-k ((n -l)θ) (n -k)! = 1 2πi C e (n-l)θφ(z) z n-k+1 dz = 1 2πi C z k-1 e -lθφ(z) e n(θφ(z)-log z) dz = 1 2πi C h(z)e ng(z) dz,
where h(z) := z k-1 e -lθφ(z) and g(z) := θφ(z) -log z. Note that g (z) = θφ (z) -1/z and that g (z) = θφ (z) + 1/z 2 . In particular, g has a single real zero at the point z(θ) solving the equation θz(θ)φ (z(θ)) = 1. Note that g (z) > 0 for all z ∈ R with |z| < r. In order to derive the asymptotics of the integral C h(z)e ng(z) dz we use the saddle point method (see, for example, [START_REF] Bruijn | Asymptotic Methods in Analysis[END_REF] or [START_REF] Flajolet | Analytic Combinatorics[END_REF] for general references) and choose the contour C to be the circle around the origin with radius z(θ) such that it passes through the zero z(θ) of the derivative g . Note that g (z(θ)) = 0, so g(z(θ)e it ) has Taylor expansion

g(z(θ)e it ) = ∞ j=0 g (j) (z(θ)) j! (z(θ)) j (e it -1) j = g(z(θ)) + g (z(θ)) 2 (z(θ)) 2 (e it -1) 2 + g (z(θ)) 3! (z(θ)) 3 (e it -1) 3 + O(t 4 )
leading to the expansions Re(g(z(θ)e it )) = g(z(θ)) -(z(θ)) 2 g (z(θ))t 2 /2 + O(t 4 ) and Im(g(z(θ)e it )) = O(t 3 ). The saddle point method yields the asymptotics

C h(z)e ng(z) dz ∼ i 2π ng (z(θ))
e ng(z(θ)) h(z(θ)).

Dividing this expression by 2πi and writing z instead of z(θ) for convenience yields

σ n-k ((n -l)θ) (n -k)! ∼ e ng(z) h(z) 2πng (z) = 1 √ 2πn (a(θ)) n b kl (θ)
with a(θ) := e g(z) = e θφ(z) /z and b kl (θ) := h(z)/ g (z) = z k-1 e -lθφ(z) / z -2 + θφ (z) = z k e -lθφ(z) / 1 + θz 2 φ (z). Thus, (18) is established. Note that for k = l = 0 we have

σ n (nθ) n! = 1 2πi C e ng(z) z dz = 1 2πi π -π
e ng(z(θ)e it ) z(θ)e it iz(θ)e it dt = 1 2π

π -π e ng(z(θ)e it ) dt.

Taking the real part yields

σ n (nθ) n! = 1 2π π -π
Re(e ng(z(θ)e it ) ) dt ∼ 1 2π

π -π e nRe(g(z(θ)e it )) dt, (20)
where the last asymptotics is based on the Laplace method as follows. Choose a sequence (δ n ) n∈N of positive real numbers satisfying nδ 2 n → ∞ and nδ 3 n → 0, for example, δ n := n -α for some fixed α ∈ (1/3, 1/2). Decomposing the first integral in (20) into the two parts

I 1 := δn -δn
Re(e ng(z(θ)e it ) ) dt and I 2 := {δn<|t|≤π} Re(e ng(z(θ)e it ) ) dt we can approximate I 1 and show that I 2 is negligible (in comparison to I 1 ) for large n. Obviously, 1 -x 2 /2 ≤ cos x ≤ 1 for all x ∈ R. Choosing x := nIm(g(z(θ)e it )) and using Im(g(z(θ)e it )) = O(t 3 ) and nt 3 → 0 as n → ∞ uniformly for all |t| ≤ δ n it follows that lim n→∞ sup |t|≤δn | cos(nIm(g(z(θ)e it ))) -1| = 0. Thus, as n → ∞, the map t → cos(nIm(g(z(θ)e it ))) converges uniformly on [-δ n , δ n ] to the constant map t → 1, which implies that

I 1 = δn -δn cos(nIm(g(z(θ)e it )
))e nRe(g(z(θ)e it )) dt ∼ δn -δn e nRe(g(z(θ)e it )) dt.

The second integral I 2 is negligible (in comparison to I 1 ) since

|I 2 | ≤ {δn<|t|≤π} e nRe(g(z(θ)e it )) dt, Re(g(z(θ)e it )) = g(z(θ)) -(z(θ)) 2 g (z(θ))t 2 /2 + O(t 4 ) and nt 2 ≥ nδ 2
n → ∞ uniformly for |t| > δ n . Thus, (20) is established. For all N ∈ N and all k ∈ N 0 it follows from (18) that

P(µ N,1 = k) = N k σ k (θ) σ N -k ((N -1)θ) σ N (N θ) = σ k (θ) k! σ N -k ((N -l)θ) (N -k)! σ N (N θ) N ! ∼ σ k (θ) k! (a(θ)) n b k1 (θ)/ √ 2πn (a(θ)) n b 00 (θ)/ √ 2πn = σ k (θ) k! b k1 (θ) b 00 (θ) = σ k (θ) (z(θ)) k k! e -θφ(z(θ)) .
Thus, µ N,1 → X in distribution as N → ∞, where X has distribution (17). In the following, for arbitrary but fixed p > 0, the convergence E(µ p N,1 ) → E(X p ) as N → ∞ of the p-th moments is established. For all k, l ∈ N 0 and all N ∈ N we have

σ N -k ((N -l)θ) (N -k)! ≤ σ N -k (N θ) (N -k)! = 1 2πi C z k-1 e N g(z) dz = 1 2πi π -π (z(θ)e it ) k-1 e N g(z(θ)e it ) iz(θ)e it dt = (z(θ)) k 2π π -π
e ikt e N g(z(θ)e it ) dt.

Taking the complex absolute value it follows for all k, l ∈ N and all N ∈ N that

σ N -k ((N -l)θ) (N -k)! ≤ (z(θ)) k 2π π -π |e N g(z(θ)e it ) | dt = (z(θ)) k 2π π -π
e N Re(g(z(θ)e it )) dt.

Since, by (20), (2π) -1 π -π e N Re(g(z(θ)e it )) dt ∼ σ N (N θ)/N !, it follows that there exists a constant N 0 ∈ N (which may depend on θ and φ but not on k and l) such that

σ N -k ((N -l)θ) (N -k)! ≤ 2(z(θ)) k σ N (N θ) N !
for all k, l ∈ N 0 and all N ≥ N 0 . In particular, for all N ≥ N 0 and all k ∈ N 0 ,

P(µ N,1 = k) = σ k (θ) k! σ N -k ((N -1)θ) (N -k)! σ N (N θ) N ! ≤ σ k (θ) k! 2(z(θ)) k = κ(θ)P(X = k),
where κ(θ) := 2e θφ(z(θ)) ∈ (0, ∞). For arbitrary but fixed p > 0 the map k → k p P(X = k), k ∈ N 0 , is integrable with respect to the counting measure ε N0 on N 0 , since

k p P(X = k)ε N0 (dk) = ∞ k=0 k p P(X = k) = E(X p ) < ∞.
By dominated convergence we therefore have convergence E(µ p N,1 ) → E(X p ) of all moments. In particular, (N -

1)c N = E((µ N,1 ) 2 ) → E((X) 2 ) > 0 and (N -1)(N -2)d N = E((µ N,1 ) 3 ) → E((X) 3 ) as N → ∞. Thus, d N /c N = O(1/N ) → 0 as N → ∞,
which ensures (see [START_REF] Möhle | A classification of coalescent processes for haploid exchangeable population models[END_REF] or [8, Theorem 4 (b)]) that the considered symmetric compound Poisson model is in the domain of attraction of the Kingman coalescent.

4 Examples

We start with the two most popular examples, the Wright-Fisher model and the Dirichlet model. Note that (7) holds for these two models. These two examples have the advantage that most calculations can be done explicitly. For example, we will verify the asymptotic results stated in Theorem 3.2 directly.

(θ) = θ k , k ∈ N 0 , θ ∈ (0, ∞). For k ∈ N 0 and θ ∈ (0, ∞) it follows that P(µ N,1 = k) = N k σ k (θ) σ N -k ((N -1)θ) σ N (N θ) = N k 1 N k 1 - 1 N N -k .
Thus, µ N,1 has a binomial distribution with parameters N and 1/N . In particular, for arbi-

trary but fixed p ∈ N, E((µ N,1 ) p ) = (N ) p /N p → 1 = E((X) p ) as N → ∞,
(θ) = [θ] k := θ(θ + 1) • • • (θ + k -1) = Γ(k + θ)/Γ(θ), k ∈ N 0 , θ ∈ (0, ∞).
Thus, for k ∈ N 0 and θ ∈ (0, ∞),

P(µ N,1 = k) = N k [θ] k [(N -1)θ] N -k [N θ] N = [θ] k k! (N ) k Γ(N θ -θ + N -k)Γ(N θ) Γ(N θ -θ)Γ(N θ + N ) .
Since Γ(x + c) ∼ x c Γ(x) as x → ∞ for any c ∈ R, it follows that

P(µ N,1 = k) ∼ [θ] k k! (N ) k (N θ + N ) -θ-k Γ(N θ + N )Γ(N θ) (N θ) -θ Γ(N θ)Γ(N θ + N ) = [θ] k k! (N ) k (N θ + N ) k θ + 1 θ -θ → [θ] k k! 1 θ + 1 k θ θ + 1 θ .
Thus µ N,1 → X in distribution as N → ∞ where X has a negative binomial distribution with parameters θ and θ/(θ + 1) ∈ (0, 1). Moreover, for arbitrary but fixed p ∈ N, E((µ N,1

) p ) = (N ) p [θ] p /[N θ] p → [θ] p /θ p = E((X) p ) as N → ∞. Thus, E(µ p N,1
) → E(X p ) as N → ∞ for all p ∈ N in agreement with the general results derived in the proof of Theorem 3.2. The solution z(θ) of the equation θz(θ)φ (z(θ)) = 1 is z(θ) = 1/(θ + 1). Note that (18) holds with a(θ) := (θ + 1) θ+1 /θ θ and b kl (θ) := (θ/(θ + 1)) lθ+1/2 (1/(θ + 1)) k , k, l ∈ N 0 , θ ∈ (0, ∞). The symmetric Dirichlet model has effective population size N e = 1/c N = (N -1)/E((µ N,1 ) 2 ) = (N θ + 1)/(θ + 1) ∼ N with = θ/(θ + 1) = 1/E((X) 2 ), in agreement with Theorem 3.2.

In the following examples are studied which do not satisfy [START_REF] Huillet | Population genetics models with skewed fertilities: a forward and backward analysis[END_REF]. We start with a model which involves the absolute Lah numbers. This example was the starting point for writing this erratum, since we recognized that the formula [START_REF] Graham | Concrete Mathematics. A Foundation for Computer Science, Second Edition[END_REF], and, hence, the relation [START_REF] Huillet | Population genetics models with skewed fertilities: a forward and backward analysis[END_REF], are not satisfied for this model.

= 1/(1 + θz 2 φ (z)) = (1 -z)/(1 + z) < 1, since 1 + θz 2 φ (z) = 1 + θz 2 φ (z) 2 1 -z = 1 + 2z 1 -z = 1 + z 1 -z .
Note that (18) holds with a(θ) = e θφ(z) /z = e θz/(1-z) /z = e 1-z /z and b kl (θ

) := z k e -lθφ(z) / 1 + θz 2 φ (z) = z k e -l(1-z) / (1 + z)/(1 -z), k, l ∈ N 0 , θ ∈ (0, ∞).
Let us generalize Example 4.3 as follows.

φ m = (m -1)! am m -1 b m-1 , m ∈ N,
where it is assumed that the real parameters a and b are either both negative or b > 0 and a ≥ 1 (such that all the coefficients φ m are nonnegative). Note that [1, Eq. ( 43)] the power series φ(z) = ∞ m=1 φ m z m /m! is the solution of the functional equation φ(z) = zf (φ(z)) with f (x) := (1 + bx) a and that φ is related to the generalized binomial series B a (see, for example, [6, p. 200]) via φ(z) = (B a (bz) -1)/b. It is readily checked that φ has radius of convergence

r := lim m→∞ (m + 1)φ m φ m+1 = 1 ab 1 - 1 a a-1 ∈ (0, ∞)
and that φ(r-) = 1/(b(a -1)) (= ∞ for a = 1), since, for a = 1, φ(r-) = 1/(b(a -1)) is the only positive solution of the equation φ(r-) = rf (φ(r-)). Differentiating both sides of the functional equation φ(z) = zf (φ(z)) yields the derivatives Thus, z = z(θ) satisfies θzφ (z) = 1. Theorem 3.2 is therefore applicable. Straightforward computations show that the limiting random variable X in Theorem 3.2 has second factorial moment E((X) 2 ) = 1 + θz 2 φ (z) = (a -x 2 )/(a(1 -x) 2 ). By Theorem 3.2, the model is in the domain of attraction of the Kingman coalescent and the effective population size N e satisfies N e ∼ N as N → ∞ with = 1/E((X) 2 ) = (a(1 -x) 2 )/(a -x 2 ) ∈ (0, 1). Note that the asymptotics (18) holds with a(θ) = e θφ(z) /z = e 1-x /z and b kl (θ) = z k e -lθφ(z)

φ (z) = f (φ(z)) 1 -zf (φ(z)) and φ (z) = 2f (φ(z))φ (z) + zf (φ(z))(φ (z)) 2 1 -zf (φ(z)) , |z| < r.
1 + θz 2 φ (z) = z k e -l(1-x) a(1 -x) 2 a -x 2 If α ≤ 2, then φ (1-) = ∞, so Theorem 3.2 is applicable for all θ ∈ (0, ∞). There seems to be no closed expression available for the solution z = z(θ) of the equation zφ (z) = 1/θ. The case α = 2 is a nice exception where it is easily seen that z(θ) = 1 -e -1/θ , θ ∈ (0, ∞). If α > 2, then φ (1-) = ζ(α -1) < ∞. In this case the equation θzφ (z) = 1 admits a solution z = z(θ) ∈ (0, 1) if and only if θ > θ c with critical value θ c := 1/φ (1-) < 1. Thus, Theorem 3.2 is not applicable for θ ≤ θ c . We leave it open for future work to discuss the limiting behavior of the random variable µ N,1 in this case.

)

  Moreover, the associated symmetric compound Poisson model is in the domain of attraction of the Kingman coalescent in the sense of [7, Definition 2.1 (a)]. The effective population size N e := 1/c N satisfies N e ∼ N as N → ∞, where

Example 4 . 1 (

 41 Wright-Fisher model) For the standard symmetric Wright-Fisher model, φ(z) = z. Therefore, ξ n has a Poisson distribution with parameter θz and σ k

  where X has a Poisson distribution with parameter 1. The solution z(θ) of the equation θz(θ)φ (z(θ)) = 1 is z(θ) = 1/θ. Note that (18) holds with a(θ) := eθ and b kl (θ) := θ -k e -l , k, l ∈ N 0 , θ ∈ (0, ∞). The effective population size N e = 1/c N = N coincides with the actual population size N , a well known result.

Example 4 . 2 (

 42 Dirichlet model) For the symmetric Dirichlet model, φ(z) = -log(1 -z), |z| < 1. Therefore, ξ 1 has a negative binomial distribution with parameters θ and 1 -z. In particular, E(ξ 1 ) = θzφ (z) = θz/(1 -z). Moreover, σ k

Example 4 . 3

 43 Suppose that φ m = m! for all m ∈ N or, equivalently, that φ(z) = z/(1 -z), |z| < 1. Note that φ (z) = 1/(1 -z) 2 and that φ (z) = 2/(1 -z) 3 = 2φ (z)/(1 -z), |z| < 1. Then (Comtet [4, Section 3.3, p. 135, Theorem B]), σ n (θ) = n k=1 L(n, k)θ k , n ∈ N, θ ∈ R, where L(n, k) := B nk (1!, 2!, . . .) = n!(n-1)!/(k!(k-1)!(n-k)!), n ∈ N, k ∈ {1, . . . , n}, denote the absolute Lah numbers. The solution z = z(θ) ∈ (0, 1) of the equation 1 = θzφ (z) = θz/(1 -z) 2 is z(θ) = 1 + θ/2 -θ(θ + 4)/2. By Theorem 3.2, the model is in the domain of attraction of the Kingman coalescent and the effective population size N e satisfies N e ∼ N with :

Example 4 . 4

 44 Suppose that (see [1, p. 402, Eqs. (50) and (51)])

For a = 1 (

 1 and b > 0) we have φ m = m!b m-1 , m ∈ N, and φ(z) = z/(1 -bz), |z| < 1/b. For a = b = 1 we are back in the previous example involving the absolute Lah numbers. For a = 2 we have φ(z) = 4z/(1 + √ 1 -4bz) 2 , |z| < 1/(4b). For a = -1 (and b < 0) we have φ m = (2m -2)!/(m -1)!(-b) m-1 , m ∈ N, and φ(z) = ( √ 1 + 4bz -1)/(2b) = 2z/(1 + √ 1 + 4bz), |z| < 1/(-4b). For general parameters a and b, to the best of the authors knowledge, there seems to be no special function related to the power series φ. Nevertheless, an explicit expression for the solution z = z(θ) of the equation θzφ (z) = 1 is obtained as follows. Let x = x(θ) be the solution in the open interval (0, 1) of the quadratic equation θx = b(1 -x)(a -x), i.e. x(θ) = θ + b + ab -(θ + b + ab) 2 -

  with x = x(θ) and z = z(θ) defined in (21) and(22). At first glance the solution x(θ) of the quadratic equation θx = b(1 -x)(a -x) seems to come 'from nowhere'. In the following an intuitive argument is provided showing how x(θ) comes into play. It is known [1, Eq. (51)] that σ n (θ) = n j=1 B(n, j)θ j with n -j + 1)(nθ) n-j+1 B(n, n -j)(nθ) n-j = nθj (n -j)(an -j + 1)bExample 4.8 (Polylog model) Fix α ∈ (0, ∞) and suppose that φ is the polylog function, i.e. φ(z) := ∞ m=1 m -α z m , |z| < 1. For α = 1 the polylog model coincides with the Dirichlet model (Example 4.2). For α → 0 we are back in Example 4.3 whereas for α → ∞ we approach the Wright-Fisher model (Example 4.1).

  are as well correct, since the Kimura model does not belong to the compound Poisson class. Thus, only [7, Section 4] from [7, Proposition 4.2] on needs correction. From the results of the previous Section 2 it follows that [7, Proposition 4.2] and [7, Theorem 4.3] and the remarks thereafter are valid for generalized Wright-Fisher models and generalized Dirichlet models with power series φ as given at the end of Section 2. However, for general compound Poisson models, Proposition 4.2 and Theorem 4.3 of [7] and the remark thereafter are wrong. Proposition 4.2 of [7] should be replaced by the following proposition. Proposition 3.1 If, for each n ∈ N, the random variable ξ n has a pgf of the form [7, Eq. (

Laboratoire de Physique Théorique et Modélisation, CNRS-UMR 8089 et Université de Cergy-Pontoise,

Avenue Adolphe Chauvin, 95302 Cergy-Pontoise, France, E-mail: thierry.huillet@u-cergy.fr 2 Mathematisches Institut, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany, E-mail: martin.moehle@uni-tuebingen.de

Acknowledgement. The authors thank Elmar Teufl for fruitful discussions concerning the Laplace method and the saddle point method.

Intuitively, when n is large, the contribution to the sum σ n (nθ) = n-1 j=0 B(n, n -j)(nθ) n-j is essentially entirely originated from indices j having the property that j/n belongs to a (small) neighborhood of x(θ). The choice of the neighborhood is rather unimportant. It is hence not surprising that x(θ) plays a crucial role in finding the asymptotics of σ n (nθ). Since the Bell numbers (23) are known explicitly, one may carry out the Laplace method in detail leading to an alternative proof of (18). We leave the details to the reader.

Closed forms for the positive real solution z = z(θ) of the equation θzφ (z) = 1 are only available for particular values of the parameter α. For example, z(θ) = 2θ -2 ( √ θ 2 + 1 -1) for α = 1/2. By Theorem 3.2, the model is in the domain of attraction of the Kingman coalescent and the effective population size N e satisfies N e ∼ N with := 1/(1+θz

which corrects the wrong asymptotic result for N e provided at the bottom of p. 538 of [START_REF] Huillet | Population genetics models with skewed fertilities: a forward and backward analysis[END_REF].

which again corrects the wrong asymptotic result for N e provided on top of p. 539 of [START_REF] Huillet | Population genetics models with skewed fertilities: a forward and backward analysis[END_REF].

All examples considered so far satisfy φ (r-) = ∞ such that for all θ ∈ (0, ∞) the equation θzφ (z) = 1 has a unique real solution z(θ) ∈ (0, r). We finally provide an example satisfying φ (r-) < ∞.