S Bereux

B Delmas

F Cazals

Boosting the analysis of protein interfaces with Multiple Interface String Alignments: illustration on the spikes of coronaviruses

Keywords: protein interface, multiple sequence alignments, interface comparison, interface RMSD, Structural Bioinformatics Library, coronaviruses, spikes, fusion proteins. Running Multiple Interface String Alignments

We introduce Multiple Interface String Alignment (MISA), a visualization tool to display coherently various sequence and structure based statistics at protein-protein interfaces (SSE elements, buried surface area, ∆ASA, B factor values, etc). The amino-acids supporting these annotations are obtained from Voronoi interface models. The benefit of MISA is to collate annotated sequences of (homologous) chains found in different biological contexts i.e. bound with different partners or unbound. The aggregated views MISA/SSE, MISA/BSA, MISA/∆ASA etc make it trivial to identify commonalities and differences between chains, to infer key interface residues, and to understand where conformational changes occur upon binding. As such, they should prove of key relevance for knowledge based annotations of protein databases such as the Protein Data Bank.

Illustrations are provided on the receptor binding domain (RBD) of coronaviruses, in complex with their cognate partner or (neutralizing) antibodies. MISA computed with a minimal number of structures complement and enrich findings previously reported.

Introduction

Protein complexes and their interface. Understanding the stability and the specificity of protein interactions is a fundamental problem [START_REF] Janin | Protein-protein interaction and quaternary structure[END_REF], both to unveil complex biological mechanisms and design therapeutics. Quantitative insights on such problems call for analysis falling in the realm of thermodynamics and kinetics. On the thermodynamics side, the strength of an interaction is given by the binding affinity, measured say by the dissociation constant K d [START_REF] Kuriyan | The molecules of life: Physical and chemical principles[END_REF]. The affinity is a subtle thermodynamic quantity, encompassing an enthalpic term derived from the interaction potential energy, and an entropic term related to the entropic penalty upon binding -see e.g. [START_REF] Rajamani | Anchor residues in proteinprotein interactions[END_REF][START_REF] Chia-En | Ligand configurational entropy and protein binding[END_REF]. From the theoretical standpoint, the binding affinity can be defined as the ratio of three so-called partition functions [START_REF] Zhou | Theory of free energy and entropy in noncovalent binding[END_REF]. These partition functions are themselves defined from the potential energy of the system, which in particular encodes interaction energies at play between atoms (electrostatic interactions, non covalent interactions, bonded interactions). This observation does not settle the question though, as the huge dimensionality of the system involving the biomolecules and the solvent precludes calculations, except maybe for small systems using massive calculations [START_REF] Shaw | Atomic-level characterization of the structural dynamics of proteins[END_REF][START_REF] Shi Chen | The dynamic conformational landscape of the protein methyltransferase SETD8[END_REF]. Application-wise, an illustrative example of the shear difficulty of obtaining accurate affinity estimates is provided by the affinity maturation of antibodies in the immune response [START_REF] Schmidt | Preconfiguration of the antigen-binding site during affinity maturation of a broadly neutralizing influenza virus antibody[END_REF]. As illustrated by this case study, the same (backbone) bound geometry may indeed corresponding to dissociation constants differing by two orders of magnitude. On the kinetics side, the residence time (time spent by a ligand in contact with its biological target) is 1/K off [START_REF] Corzo | Time, the forgotten dimension of ligand binding teaching[END_REF][START_REF] Bernetti | Kinetics of drug binding and residence time[END_REF]. Estimating reaction rates (K on and K off ) is also extremely difficult. This state of affairs partly explains the success of models based on geometric proxys (distances, surfaces and volumes) to estimate structural, thermodynamic, and kinetic properties. Such models originate with the seminal work of Richards for individual molecules [START_REF] Richards | The interpretation of protein structures: total volume, group volume distributions and packing density[END_REF] and Janin for complexes [START_REF] Chothia | Principles of protein-protein recognition[END_REF]. They proved instrumental to understand the specificity and affinity of interactions [START_REF] Janin | Protein-protein interaction and quaternary structure[END_REF]; in some cases, they even proved good enough to estimate binding energies up to (of the order of) 1.4kcal/mol [START_REF] Marillet | High resolution crystal structures leverage protein binding affinity predictions[END_REF]. To study two partners forming a complex, such models require identifying the atoms in contact. To this end, Voronoi models, be their affine [START_REF] Cazals | Revisiting the Voronoi description of protein-protein interfaces[END_REF] or curved [START_REF] Olechnovič | Voronota: a fast and reliable tool for computing the vertices of the voronoi diagram of atomic balls[END_REF], provide a parameter free approach to model interfaces, and affine models in particular give a direct access to various structural parameters including the number of interface atoms and residues, the buried surface area, the number of binding patches, or the interface curvature [START_REF] Cazals | Revisiting the Voronoi description of protein-protein interfaces[END_REF].

These parameters are typically sufficient to compare interfaces for cases belonging to various protein families (protease-inhibitors, enzyme-substrate, antibody-antigen, signal transduction, etc) [START_REF] Prasad Bahadur | A dissection of specific and non-specific protein-protein interfaces[END_REF], or cases within a protein family [START_REF] Marillet | Novel structural parameters of ig-ag complexes yield a quantitative description of interaction specificity and binding affinity[END_REF]. On the other hand, in processing complexes specifically addressing a biological case, say a family of homologous proteins targeting the same cognate partner, it is also critical to study correlations between structural parameters describing interfaces and sequence related pieces of information. For sequences, the natural route in to resort to multiple sequence alignments (MSA) [START_REF] Pevsner | Bioinformatics and functional genomics[END_REF].

Multiple Interface String Alignment (MISA).

In this work, we introduce MISAs, a visualization tool to display coherently various sequence and structure based statistics at protein-protein interfaces (SSE elements, buried surface area, ∆ASA, B factor values, etc). The benefit of MISAs is to collate annotated sequences of (homologous) chains found in different biological contexts (unbound or bound with different partners).

We illustrate the interest to analyse the receptor-binding domain (RBD) or the virus SARS-Cov-2 causing the Covid19 outbreak. First, we compare the receptor-binding domains (RBD) of SARS-Cov-1 and SARS-Cov-2 (two phylogenetically-related betacoronavirus). Second, we compare the interfaces of their RBD bound to their common cell receptor and with antibodies.

The implementation of MISAs available in the SBL (http://sbl.inria.fr, [START_REF] Cazals | The Structural Bioinformatics Library: modeling in biomolecular science and beyond[END_REF]), see https://sbl. inria.fr/doc/Multiple_interface_string_alignment-user-manual.html.

2 Methods: MISAs -multiple interface string alignments

MISA

We assume a collection of crystal structures of the bound and unbound types. We call a polypeptide chain in these structures a chain instance. We also assume that a multiple sequence alignment is available for the sequences of the instances of interest. Practically, we expect two settings: (i) chain instances with identical sequence: since numberings used in the respective PDB files are identical, the identity alignment is used; (ii) homologous proteins: a multiple sequence alignment providing a coherent numbering of amino acids (a.a.) is taken for granted.

The terminology used in the sequel is as follows. We study chain instances and compare them against one another using one MISA for each so-called MISA id. A MISA is based on interface strings (i-string), which following conventions for sequence alignments, use the one-letter code of (a.a.) plus three additional symbols {_ * -}. We now detail these concepts.

MISA id. A complex C i is specified by two sorted lists of chains ids i.e. C i = ({A j }, {B j }) for the two partners; each list is endowed with a partner/structure name. Likewise, an unbound structure is specified by a sorted list of chain ids U i = {A j }, and is also given a partner/structure name (e.g. antibody, antigen, etc.) We consider a collection of complexes C = {C i }, and optionally a collection of unbound structures U = {U i }. The minimal setup is naturally that of a single complex without any unbound structure.

Our goal is to build one MISA for each so-called MISA id, which we formally define as:

Definition. 1
The MISA id of a chain in a complex or unbound structure is the string defined by the structure name followed by the index of the chain in its sorted list {A j } or {B j }. (Nb: by convention, indices start at 0.)

Note that because of the sortedness, a given chain gets the same MISA id in a complex or unbound structure.

Example 1 Consider two antibody-antigen complexes. C 1 = ({H, L}, {A}), C 2 = ({M, N }, {B}), each involving the heavy chains (chains H and M), the light chains (chains L and N), and the antigen (chains A and B). The two structure names are thus antibody (IG for short) and antigen (Ag for short). The MISA id of the heavy chain is IG_0, that of the light chain IG_1, and that of the antigen Ag_0.

Voronoi interface in a binary complex. The two lists of chains in the specification of a complex C i define the so-called partners in the complex, called A and B for short. In the sequel, we build the Voronoi interface for these partners, using the Voronoi (power) diagram of the solvent accessible model [START_REF] Cazals | Revisiting the Voronoi description of protein-protein interfaces[END_REF][START_REF] Janin | Protein-protein interaction and quaternary structure[END_REF][START_REF] Loriot | Modeling macro-molecular interfaces with Intervor[END_REF]. More precisely, recall that the Voronoi model identifies pairs of atoms, one on each partner, which are either directly in contact as their Voronoi cells are neighbor, or are contacting a common crystallographic water molecule. Therefore, an a.a. contributing at least one interface atom is called an interface residue/a.a.. Practically, we use the sbl-intervor-ABW-atomic.exe executable from the package Space_filling_model of the Structural Bioinformatics Library ( [START_REF] Cazals | The Structural Bioinformatics Library: modeling in biomolecular science and beyond[END_REF], http://sbl.inria.fr).

Interface strings (i-strings). In the sequel, we present MISAs informally, referring the reader to SI Section Supplemental: formal specification of the method. We represent each chain instance with a so-called interface string encoding properties of its interface amino acids. The interface string of a chain instance is a character string with one character per residue, and is actually defined from all instances with the same MISA id. To build this character string, we first build a consensus interface based on all Voronoi interfaces of all complexes in the set C. At each position of the consensus interface, the most frequent residue observed in all the bound structures is chosen as the consensus residue-with ties broken using the alphabetical order of the one letter code of a.a.. Using this consensus interface, the residues of a given chain instance (bound or unbound) are assembled into the so-called interface string:

Definition. 2 The interface string (i-string) of a chain instance is the string with one character per amino acid, defined as follows:

1. Residue not part of the consensus interface :

• Displayed with a dash "-" if it is part of the crystal structure, and underscore "_" otherwise.

Residue part of the consensus interface:

• Residue not found in the crystal structure: displayed with the star '*'.

• Residue found at the interface for this particular chain: displayed with the uppercase one letter code if the a.a. matches the consensus a.a., or with the lowercase letter otherwise. • Residue not found at the interface for this particular chain, even though the corresponding position contributes to the consensus interface (for other chains): displayed in an italicized uppercase/lowercase letter. (Note that this is the case for all residues of unbound structures-as no partner implies no interface.)

Practically, it is often sound to restrict the study to a range of amino-acids on the chains studied, whence the following:

Definition. 3 A window series for a MISA id is a sequence of integer ranges of residue ids {(B i , E i )} used to restrict the display of the i-string.

MISA. We use the MSA to align the symbols of interface strings, yielding:

Definition. [START_REF] Chia-En | Ligand configurational entropy and protein binding[END_REF] The MISA of all chain instances in C ∪ U with the same MISA id is the multiple alignment of their interface strings.

Colored MISAs

We further color the letters of a MISA to encode properties of biological/biophysical interest.

MISA/SSE: coloring based on Secondary Structure Elements. The type of SSE a given a.a. belongs to is especially useful when comparing bound and unbound structures, to assess perturbations in the hydrogen bonding network. Practically, we use the dictionary of SSE from [START_REF] Kabsch | Dictionary of protein secondary structure: pattern recognition of hydrogenbonded and geometrical features[END_REF].

MISA/BSA: coloring based on Buried Surface Area. Consider a chain instance in a complex. The BSA of this chain is defined as the accessible surface area (ASA) [START_REF] Richards | The interpretation of protein structures: total volume, group volume distributions and packing density[END_REF] of this chain in the partner alone minus the ASA of the chain in the complex. We report the BSA on a per residue basis, computed using the algorithm from [START_REF] Cazals | Computing the volume of union of balls: a certified algorithm[END_REF].

Remark 1 The Voronoi interface model identifies privileged contacts between atoms of the partners. Interestingly, selected interface atoms, generally backbone atoms, can be buried within their own subunit [START_REF] Cazals | Revisiting the Voronoi description of protein-protein interfaces: Algorithms[END_REF]. Since such atoms do not have any BSA (they do not have ASA in their own subunit), a residue containing only such atoms is an interface residue for which the BSA information is irrelevant.

MISA/∆ ASA: coloring based on the variation of accessible surface area. A limitation of the BSA is that its calculation uses the geometry of the bound structure only. The calculation is thus oblivious to conformational changes which may be at play in case of induced fit or conformer selection. To mitigate the previous plot, we also provide the so-called ∆ ASAcoloring scheme.

Consider an interface partner P (= A or B), in the complex, and consider the i-th residue of one of its chains. Let ASA b i be the ASA of this i-th residue in the structure involving only the chains defining partner P . Also denote ASA u i the average ASA of the i-th residue in unbound structures with the same MISA id. We compute for the i-th residue of a bound structure the quantity ASA b i -ASA u i and display it with a color map.

Remark 2

The quantity ASA b i -ASA u i is undefined when the corresponding residues are absent in the unbound crystal structures (* in i-string).

MISA/B-factor: coloring based on B-factors. The B-factors reflects the atomic thermal motions. Selected recent crystal structures report this information as a 3x3 ANISOU matrix (the anisotropic Bfactor). In order to have a single quantity for all the crystals, the ANISOU matrix B anisou is converted into B-factor thanks to the formula B f actor = trace(B anisou )/3 [START_REF] Yang | Comparisons of experimental and computed protein anisotropic temperature factors[END_REF]. Optionally, one can choose to normalize the B-factor with respect to (i) all the residues in the chain, or (ii) all the displayed residues.

Availability and practical matters

MISAs are implemented within the Multiple_interface_string_alignment package of the Structural Bioinformatics Library ( [START_REF] Cazals | The Structural Bioinformatics Library: modeling in biomolecular science and beyond[END_REF], http://sbl.inria.fr), see https://sbl.inria.fr/doc/Multiple_interface_string_alignment-user-manual.html.

The calculation and the analysis of MISAs is based on four scripts-see online documentation:

• sbl-misa.py: building MISAs from a description of complexes and possibly unbound structures.

• sbl-misa-mix.py: mixing selected colored MISAs into a single (html) file.

• sbl-misa-bsa.py: displaying the BSA of all (or selected user defined) residues.

• sbl-misa-diff.py: comparing i-strings and associated properties (in particular BSA) of two interfaces.

Has two modes: comparing two i-strings; comparing one i-string and a list of interface residues typically coming from a publication.

Illustration of these scripts to investigate the questions discussed in the next section are provided in the jupyter notebook which accompanies our software.

Results: studying the receptor-binding domain (RBD) of coronaviruses

Similarly to other enveloped viruses, the SARS-Cov-2 virus enters cells using fusion proteins. Having recalled the known facts on this mechanism, we show that MISAs are helpful in two respects: (i) to automate such studies and refine some of the findings reported so far, and (ii) to provide a coherent view of several sequence-structure based statistics, which is instrumental in particular to understand the binding affinity of SARS-Cov-2 RBD for its target.

Biological problem

Coronavirus entry into cells is mediated by the so-called spike that is responsible for receptor binding and membrane fusion [START_REF] Tortorici | Structural insights into coronavirus entry[END_REF][START_REF] Harrison | Viral membrane fusion[END_REF]. The covid19 outbreak triggered an impressive number of structural studies on the entry of SARS-Cov-2, namely [START_REF] Wrapp | Cryo-em structure of the 2019-ncov spike in the prefusion conformation[END_REF] [28] [29] [START_REF] Hoffmann | Sars-cov-2 cell entry depends on ACE2 and TMPRSS2 and is blocked by a clinically proven protease inhibitor[END_REF] [31] [32] [33]. Interactions between the spikes of SARS-Cov-2 with antibodies have also studied [START_REF] Walls | Unexpected receptor functional mimicry elucidates activation of coronavirus fusion[END_REF], [START_REF] Wrapp | Structural basis for potent neutralization of betacoronaviruses by single-domain camelid antibodies[END_REF], [START_REF] Ju | Human neutralizing antibodies elicited by sars-cov-2 infection[END_REF]. These papers naturally unveiled critical aspects of this complex process, based on experiments and also sequence based and structure based analysis. Spikes are homotrimers of the protein S [START_REF] Delmas | Assembly of coronavirus spike protein into trimers and its role in epitope expression[END_REF][START_REF] Li | Structure, function, and evolution of coronavirus spike proteins[END_REF]. Each chain consists of two domains S1 and S2, separated by cleavage sites denoted S1/2 and S2' [START_REF] Walls | Tectonic conformational changes of a coronavirus spike glycoprotein promote membrane fusion[END_REF][START_REF] Walls | Structure, function, and antigenicity of the SARS-CoV-2 spike glycoprotein[END_REF][START_REF] Wrapp | Structural basis for potent neutralization of betacoronaviruses by single-domain camelid antibodies[END_REF]. Domain S1 contains the receptor-binding domain (RBD), while domain S2 contains the machinery responsible for the fusion of the virus envelope with the membrane of the cell. In short, the main steps of the mechanism are as follows [START_REF] Walls | Tectonic conformational changes of a coronavirus spike glycoprotein promote membrane fusion[END_REF][START_REF] Shang | Cell entry mechanisms of SARS-CoV-2[END_REF]: (1) Attachment of the RBD to its target thanks to the receptor binding-motif (RBM), (2) proteolysis cleavage/activation at S1/S2 removing the S1 subunit, (3) second cleavage at S2', refolding of fusion machinery -anchoring of fusion peptide into the target membrane, (4) membrane fusion and genome delivery into the cytoplasm. In human, both SARS-Cov-1 and SARS-Cov-2 targets ACE2 [START_REF] Yan | Structural basis for the recognition of SARS-CoV-2 by full-length human ACE2[END_REF], a membrane bound enzyme catalysing the hydrolysis of angiotensin II into angiotensin, as a receptor to enter into cells. The following facts are noticeable [START_REF] Shang | Cell entry mechanisms of SARS-CoV-2[END_REF]. The virulence of SARS-Cov-2 owes in particular to the presence of a polybasic cleavage site (insertion of three R) collated to the usual S1/S2 cleavage site. The RBD of SARS-Cov-2 stands up less often than that of SARS-Cov-1, which likely favors evasion from the immune system. (Nb: the two conformations are called up/down or closed/open.) This also disfavors binding to the target (binding in the down configuration yields steric clashes), but the lesser proportion of RBD standing up is rescued by a higher binding affinity for ACE2 [START_REF] Shang | Cell entry mechanisms of SARS-CoV-2[END_REF]. Interestingly, a phylogenetically distant human alphacoronavirus, HCoV-NL63 also targets ACE2 as a receptor.

In the following, we illustrate the ability of interface string and MISAs to shed light on these complex facts. In doing so, our goal is a presentation of the merits of MISAs, rather than an arbitration of the merits of different publications which investigated the same systems. For this reason, we use a minimal number of crystal structures to get informative MISAs, namely two bound and two unbound structures, choosing those with the highest resolution. As we shall see, even such a simple study sheds new light on a number of published facts.

Comparing the RBD of SARS-Cov-1 and SARS-Cov-2 in complex with ACE2

Baseline. The spikes of SARS-Cov-1 and SARS-Cov-2 are highly homologous proteins binding ACE2.

Pairwise alignment shows that the overall percent protein sequence identity is about 76% [START_REF] Jaimes | Phylogenetic analysis and structural modeling of SARS-CoV-2 spike protein reveals an evolutionary distinct and proteolytically-sensitive activation loop[END_REF]. The S1 subunit is less conserved (64% identity) than S2 (90% identity). Within S1, the receptor-binding domain (RBD) is 74% identical. The initial analysis of the complex between the RBM of SARS-Cov-1 and its target ACE2 was first carried out in [START_REF] Li | Structure of SARS coronavirus spike receptor-binding domain complexed with receptor[END_REF]. In this work, it is noted that the BSA at the interface is ∼ 1700Å . These recent publications take the 14 a.a. identified in [START_REF] Li | Structure of SARS coronavirus spike receptor-binding domain complexed with receptor[END_REF] for granted. Note that in furthering these analysis, we sometimes refer to the jupyter notebook which accompanies our software, and propose selected analysis using the four scripts of the Multiple_interface_string_alignment package.

Whole Voronoi interfaces. The Voronoi interface calculation yields comparable results (Figs. 1 and2). For SARS-Cov-1 (PDBid: 2ajf, RBD: chain E, ACE2: chain A), one obtains 246 interface atoms making up 2 binding patches, and defining overall 414 pairwise contacts. For SARS-Cov-2 (PDBid 6m0j, RBD: chain E, ACE2: chain A), one gets 212 interface atoms also disposed on two binding patches, and defining overall 371 pairwise contacts. The two interface patches correspond to the transition between a beta sheet and loops on the RBD (Fig. 1(B,C)), dotted line-segments). Moving to interface strings (Fig. 2, we note that the RBD contribution is is characterized by five main stretches along the backbone. Interestingly, two stretches (yellow and red) span the whole interface, crossing on top of one another in a X like figure (Fig. 1(C)).

Buried surface area. The BSA is essentially proportional to the number of interface atoms [START_REF] Janin | Protein-protein interaction and quaternary structure[END_REF], and is also a key parameter to estimate binding affinities [START_REF] Janin | A minimal model of protein-protein binding affinities[END_REF][START_REF] Marillet | High resolution crystal structures leverage protein binding affinity predictions[END_REF].

We first inspect BSA values obtained for two complexes in each case (Fig. S1 and Fig. S2), and compare them against the aforementioned 1700Å 2 : SARS-Cov-1: BSA(2ajf chains E+A)=925+888=1813Å 2 ; BSA(2ajf, chains F+B)=864+817=1681 Å 2 ; SARS-Cov-2: BSA(6m0j chains E+A)=887+843=1730Å 2 ; BSA(6lzg, chains B+A)=1120+1098=2218Å 2 ). While the values are coherent for SARS-Cov-1, it appears that the complex found in 6lzg features 30% more BSA. Inspecting the number of interface atoms readily provides the explanation (output of sbl-intervor-ABW-atomic.exe, see jupyter notebook): for 6lzg, one get 156 interface atoms for the RBD, 164 interface atoms for ACE2, and 22 water molecules sandwiched between the partners; these numbers drop to 101, 110 and 1 respectively for 6m0j. This reminds us a well known fact, namely that solvation by water molecules with low temperature factors (80 units in our case) may significantly change the interface model [START_REF] Cazals | Revisiting the Voronoi description of protein-protein interfaces[END_REF].

Beyond these global values, MISA/BSA gather BSA values for all atoms of the same a.a. (Fig. S1). Two interesting facts emerge. First, both for SARS-Cov-1 and SARS-Cov-2, the residues which bury the largest ASA are found at both endpoints of the fifth stretch (Fig. S1), contributing to the stabilization of the interface. For SARS-Cov-1, one notes in particular: Y475, T486, Y491; and for SARS-Cov-2: F486, Y489, T500, Y505. We also note that some of these a.a. bury a great deal of SAS (Fig. S2), in particular Y475 for SARS-Cov-1, and Y489 for SARS-Cov-2.

Interface composition. We proceed with the inspection of the number of interface a.a., using the 14 a.a. reported in [START_REF] Li | Structure of SARS coronavirus spike receptor-binding domain complexed with receptor[END_REF] as baseline.

Focusing on the RBD of SARS-Cov-1, we note that this number is precisely 29 for the crystal structures used (Fig. 2, MISA/SSE). It should be stressed that the this is not an artifact since two atoms are in contact in a Voronoi model provided that their Voronoi restrictions intersect. (Nb: the restriction of a ball is the intersection of the atomic ball and the Voronoi/power region.) We note in particular that three residues omitted from [START_REF] Li | Structure of SARS coronavirus spike receptor-binding domain complexed with receptor[END_REF] have BSA larger than 30 Å 2 in (2ajf, chain E): L443: 34.93 Å 2 , P462: 49.79 Å 2 , and I489 : 32.87 Å 2 (jupyter notebook).

Switching to SARS-Cov-2, these numbers are 27 and 39 respectively, for the aforementioned two crystal structures-a discrepancy owing to solvation by structural water. Letting these count alone, a merit of MISAs appears immediately: as opposed to previous presentations, which single out the 14 a.a. of interest in the whole RBM sequence, it appears immediately that the interface a.a. are located within five main stretches along the sequence.

Interface strings in MISAs also single out the relatively high abundance of tyrosine (Y), and glycine (G) amongst interacting residues. Such a.a. have been shown to cooperatively account for specific recognition in particular in antibody-antigen interactions [START_REF] Birtalan | The intrinsic contributions of tyrosine, serine, glycine and arginine to the affinity and specificity of antibodies[END_REF]. The role of tyrosines at interfaces is well documented [START_REF] Koide | The importance of being tyrosine: lessons in molecular recognition from minimalist synthetic binding proteins[END_REF], both in terms on interaction types (non polar, cation-pi and H bonds) and dynamics (the rigid side chain entails a lesser entropic penalty upon binding). Again, this is not visible from previous representations [31, SI Fig 1], [START_REF] Hoffmann | Sars-cov-2 cell entry depends on ACE2 and TMPRSS2 and is blocked by a clinically proven protease inhibitor[END_REF]Fig 2B], and [START_REF] Walls | Structure, function, and antigenicity of the SARS-CoV-2 spike glycoprotein[END_REF]Fig 2C] since the interface amino acids are not identified in a systematic manner.

Conformational changes and binding affinity. MISA/SSE show that the RBD of SARS-Cov-2 is much less structured than that of SARS-Cov-1: while all residues are resolved in the unbound structures of SARS-Cov-1, a total of 12 of them are not so in the unbound structures of SARS-Cov-2 processed here (* in the i-strings, Fig. 2). The a.a. concerned are nonpolar (AGGF) or uncharged polar (NSY). Hydrophobic patches on protein surfaces are notorious and contribute to the recognition of cognate substrates [START_REF] Lijnzaad | Hydrophobic patches on the surfaces of protein structures[END_REF]. The balance observed here hints at a rather hydrophobic patch also preserving the ability to make hydrogen bonds via NSY. The color changes in MISA/SSE convey the same information: more frequent color changes for SARS-Cov-2 show that binding triggers more significant changes of hydrogen bonding patterns.

The analysis of MISA/∆ ASAcomplements this view (Fig. S2). For SARS-Cov-1, the interface a.a. have a balanced behavior, some gaining some loosing ASA, in the mild range (-10..+30Å 2 ). For SARS-Cov-2 on the other hand, the scale is shifted towards negative values (< -150.. + 35Å 2 ). In other words, the binding entails a compactification of the RBD in SARS-Cov-2.

Speaking of conformational changes, we further our analysis with a clustering of structures in terms of lRMSD, which is of special interest to assess the conformational changes between the up and down conformation of the RBD. In this case, no significant difference is observed in terms of lRMSD computed from all backbone atoms (Fig. S3). Phrased differently, MISAs show subtle details at the individual a.a. level, which are overlooked by global lRMSD calculation.

Concluding, the previous discussion shed lights on the determinants of the affinity of the RBDs of SARS-Cov-1 and SARS-Cov-2 for their receptors [START_REF] Shang | Cell entry mechanisms of SARS-CoV-2[END_REF]. For SARS-Cov-2, the unstructured nature hints at a large entropic penalty upon binding. This penalty is clearly counterbalanced by a strong enthalpic component in the dissociation free energy, which we see in particular from the BSA.

Interactions between the RBD of SARS-Cov-2 and antibodies

Neutralizing antibodies targeting the spikes of viruses can implement various strategies [START_REF] Corti | Broadly neutralizing antiviral antibodies[END_REF], including (i) masking the RBD, (ii) preventing the conformational changes triggering the fusion envelope-membrane, or may be more surprisingly by (iii) mimicking the receptor and triggering early conformational changes of the fusion machinery far away from the cell membrane [START_REF] Walls | Unexpected receptor functional mimicry elucidates activation of coronavirus fusion[END_REF]. We illustrate again the interest of MISAs in this context.

We first consider the complex (SARS-Cov-2-RBD, IG CR3022), the latter being an IG isolated from a convalescent SARS patient. Since this IG also targets the RBD, two critical questions are investigated in [START_REF] Yuan | A highly conserved cryptic epitope in the receptor binding domains of SARS-CoV-2 and SARS-CoV[END_REF]: first, the comparison of the complexes complexes SARS-Cov-1-RBD with CR3022 and SARS-Cov-2-RBD with CR3022, so as to understand in particular the lesser binding affinity of CR3022 for SARS-Cov-2-RBD (Kd=115nM versus Kd=1nM); second, the absence of in vitro neutralization of SARS-Cov-2 by CR3022, taking into in particular the up/down conformations of the RBD and the role of glycosylation.

Interestingly, the first point use MISAs prepared by hand [28, Fig. 2A]. The systematic nature our automatically generated MISA/SSE shows modest albeit interesting refinements. First, one sees immediately that the regions of the RBD targeted by ACE2 and CR3022 are disjoint (Figs. 3 and4; jupyter notebook). Second, using the two complexes RBD-CR3022 in the asymmetric unit of the crystal, we note that the interface involves 33 rather than 28 a.a. which bury surface area. Those omitted from [28, Fig. 2A] are sitting on the boundary of the interface though (Fig. 3(B, VdW balls)). MISA/BSA and MISA/∆ ASAalso immediately identify those a.a. with the largest contribution to BSA (Fig. S4 and Fig. S5). Interestingly too, we note that binding with CR3022 triggers significant conformational changes with significant loss (Y380) or gain (H519) of ASA, as evidence by MISA/∆ ASA.

We also compare the interface (SARS-Cov-2-RBD, ACE2) against that of (SARS-Cov-2-RBD, P2B-2F6) [START_REF] Ju | Human neutralizing antibodies elicited by sars-cov-2 infection[END_REF]. With 10 shared residues, 10 exclusive residues for RBD -P2B-2F6, and 29 exclusive residues for RBD -ACE2, this comparison shows a clear competition between the two binders (jupyter notebook). A direct comparison against the 12 residues reported in [START_REF] Ju | Human neutralizing antibodies elicited by sars-cov-2 infection[END_REF] also shows that 8 residues reported by the Voronoi model have been omitted (Fig. 3(C, VdW balls)). Of particular interest are R346 (BSA: 18.18Å 2 ) and T470 (BSA: 21.02Å 2 ).

To complement this analysis, we consider the miniprotein inhibitors competing with ACE2 to bind the RBD [START_REF] Cao | De novo design of picomolar SARS-CoV-2 miniprotein inhibitors[END_REF]. Two design these molecules, two strategies were used, one exploiting the α helix of ACE2, and one providing de novo binders. After several rounds of optimization and testing (binding affinity measurements, solubility tests, thermostability), two designs termed LCB1 (56 amino acids) and LCB3 (64 amino acids) were selected, both binding the RBD with K d below 1nM. The bound structures of LCB1 and LCB2 in complex with the RBD were also determined by cryo-EM. We use these structures to compare at a glance the interfaces (SARS-Cov-2-RBD, ACE2), (SARS-Cov-2-RBD, VHH72), (SARS-Cov-2-RBD, CR3022), (SARS-Cov-2-RBD, LCB1), and (SARS-Cov-2-RBD, LCB2). Remarkably, the MISA involving LCB1 and LCB3 are those which are most similar to those of ACE2 (jupyter notebook). MISA/BSA also shows a noticeable feature of the structures. Taking the example of LCB1 in complex with the RBD, the interface present in the first structure (PDBid: 7jzu) has BSA of 935.23 Å 2 , while the three interfaces present in the second structure (PDBid: 7jzl) respectively have BSA of 371 Å 2 (chains A and E, RBD down), 181 Å 2 (chains C and G, RBD up), and 198 Å 2 (chains B and F, RBD up). These discrepancies owe to the much better resolution of side chains in PDBid 7jzu (jupyter notebook; Fig. S6).

Altogether, these illustrations stress again the systematic character of MISAs and their ability to compare at a glance coarse grain models of interfaces.

Outlook

The study of biomolecular interactions benefits from experimental and theoretical methods. On the experimental side, structure determination methods provide atomic coordinates for (ensembles) of structures, while thermodynamic/kinetic studies ambition to measure binding affinities and the underlying kinetic properties. Such experiments are often conducted in conjunction with directed mutagenesis, to unveil the specific role of amino acids. On the theoretical side, structure, thermodynamics and kinetics are studied using two classes of methods, namely interaction based methods, and learning based methods. As the name suggest, interaction based methods rely on covalent and non covalent potential energies models and associated force fields. Example such interactions are salt bridges, hydrogen bonds, π stacking, etc. Such models generally have a limited spatial and time resolution, which imposes simulation methods to compute observables by means of averages over ensembles. Unfortunately, properties on long time scales are generally out of reach, so that qualitative descriptors of interaction patterns are resorted to. On the other hand, learning based method typically mix geometry based models with biochemical annotations, so as to design proxys for the key determinants of affinity and specificity. Such descriptors are known to be robust, and under suitable hypothesis, have proven good enough to estimate reliably binding affinities. As illustrated by the example of the RBD of coronaviruses, most studies performed recently tend to promote specific interactions, usually backed up experimentally. This strategy is not entirely satisfactory, and one may actually quote [START_REF] Li | Structure of SARS coronavirus spike receptor-binding domain complexed with receptor[END_REF] on the interaction (SARS-Cov-1-RBD, ACE2): "The residues singled (authors' note: the 14 a.a.) out for description in the preceding paragraphs are not, of course, the only ones critical for the tight complementarity of the SARS-CoV RBD and human (or palm civet) ACE2.". To promote a more comprehensive approach, we introduce MISAs, namely alignments between interface strings defined from geometric models, and coding biological / biophysical properties of interest. While Voronoi interface models are admittedly less precise than interaction based models, they are exhaustive/systematic, parameter free, and can be pulled back onto sequences. The encoding of interactions obtained, called interface strings, provides a concise way to capture global and local properties of complexes involving related proteins at a glance, as illustrated while analysis the RBD of coronaviruses.

We anticipate that the encoding of interfaces obtained will become a standard way to concisely represent interactions for families of homologous molecules, helping to understand protein interactions, identify critical residues, and therefore design therapeutics.

Artwork

SARS-Cov-1

Cyan: ACE2; red: RBD (PDB 2ajf)

(A) (B) (C) (C)

SARS-Cov-2

Cyan: ACE2; red: RBD (PDB 6m0j) ). These operators, as presented above, depend on two indices: i which designates the complex index and j which designates the MISA id. In the code, these operators are implemented at constant j, i.e. there is one instance of each operator by MISA id. These operators are used to compute the consensus interface, and from there to create an istring for each chain. The simultaneous display of all the istring corresponding to the same MISA id is a MISA.

(A) (B) (C)

Main steps and associated classes

Step 1. : Parsing the input Main class : class MISA_preprocessor Parse the specification file and gather the structural data sbl-misa.py starts by reading the input files, and possibly runs sbl-intervor-ABW-atomic.exe and/or sbl-vorlume-pdb.exe. (The script controls that every structure has a coherent number of chains among the different complex specifications, and if it is not the case, it runs only on the complete structures). It parses the input specification file, and infers the MISA id of each chain.

Step 2. : Constructing the MISA Main class : classes MISA_constructor, MISA Initialize the MISA by gathering the chains with the same MISA id and aligning them In the following, we assume a mapping between the original indexing of the residues in each chain, and the renumbering in the MSA. For homologous chains, this mapping may be computed ClustalOmega. The construction of the MISA involves three steps: • Constructing the consensus interface, class Interface_pool : The consensus interface is computed for each MISA id from the set of the bound chains, by taking the most frequent residue at each position.

• Constructing the i-string, class MISA : for each MISA, we compute its instance of I ij [k], by comparing the residues with the consensus interface and marking them accordingly. Only the residues which are part of a windows, if a windows was provided, are displayed.

Remark 3 Practically, each residue is represented as a triple (aa-or-hetero-code, resid, insertion code), which makes it possible to handle insertion codes. This representation is used both for the keys and values of the aforementioned mapping. The corresponding class is Residue_mapping. Step 4. : Recording the cMISA Main class : class cMISA_recorder • The third is a matrix showing the number of atoms involved in the calculation of i-RMSD for each pair of chains.

As the i-RMSD is computed for each of the three grainmodes above, these three figures are produced for each of them, and the grainmode is indicated in their title.

2

 2 , and that the interface involves 18 a.a. on the receptor side and 14 a.a. on the spike side. These analysis are extended to SARS-Cov-2 and other SARS viruses, in particular in[START_REF] Shang | Structural basis of receptor recognition by SARS-CoV-2[END_REF] SI Fig 1],[START_REF] Hoffmann | Sars-cov-2 cell entry depends on ACE2 and TMPRSS2 and is blocked by a clinically proven protease inhibitor[END_REF] Fig 2B], and[START_REF] Walls | Structure, function, and antigenicity of the SARS-CoV-2 spike glycoprotein[END_REF] Fig 2C]

Figure 1 :Figure 2 :

 12 Figure 1: SARS-Cov-1 and SARS-Cov-2: Voronoi interfaces of the RBD in complex with ACE2. (Panel A) Side view of the Voronoi interface, with contributing a.a. as new ribbons. Note in particular the many disconnected contributions of SSE of the RBD to the interface. (Panel B) Top view of the interface, with interface a.a. as ribbons, and their side chains in CPK mode. (Panel C) The four or five ranges of consecutive a.a. identified by MISA, with backbone in CPK mode, on top of the Voronoi interface. SARS-Cov-1: 390-396 (violet), 404-408 (blue), 426-443 (yellow), 460-463 (orange), 470-492 (red); SARS-Cov-2: 417-421 (blue), 439-456 (yellow), 473-477 (orange), 484-506 (red)

  (A) SARS-Cov-2-RBD (red) -ACE2 (cyan) (PDB 6m0j) (B) SARS-Cov-2-RBD (red) -CR3022 (cyan) (PDB 6yla) (C) SARS-Cov-2-RBD (red) -P2B-2F6 (cyan) (PDB 7bwj) (D) SARS-Cov-1-RBD (red) -VHH-72 (cyan) (PDB 6waq)

Figure 3 :

 3 Figure 3: SARS-Cov-2 and SARS-Cov-1: RBD bound to the various IG / VHH: Voronoi interfaces. (A) Complex with ACE2 presented for reference (A). (B) Complex from[START_REF] Yuan | A highly conserved cryptic epitope in the receptor binding domains of SARS-CoV-2 and SARS-CoV[END_REF]. Residues in VdW mode are identified by the Voronoi interface model but are not listed in[START_REF] Yuan | A highly conserved cryptic epitope in the receptor binding domains of SARS-CoV-2 and SARS-CoV[END_REF]. (C) Complex from[START_REF] Ju | Human neutralizing antibodies elicited by sars-cov-2 infection[END_REF]. Residues in VdW mode are identified by the Voronoi interface model but are not listed in[START_REF] Ju | Human neutralizing antibodies elicited by sars-cov-2 infection[END_REF].

Figure 4 :

 4 Figure 4: MISA/SSE for structures of Fig. 3. MISA of ACE2 not shown.

Figure S 6 :

 6 Figure S 6: Interfaces between LCB1 and the RBD in two different structures [47]. New cartoons: molecules; CPK representations: side chains; green polygons: Voronoi interface between the two partners. Left: PDBid 7jzu, chains A and B, BSA = 935.23 Å 2 ; Middle: PDBid 7jzl, chains A and E, BSA = 371 Å 2 ; Right: PDBid 7jzl, chains B and F, BSA = 198 Å 2 . Discrepancies in the BSA owe to a much better resolution of side chains in PDBid 7jzu.

•

  Initializating the MISA, class MISA_constructor : All the chains sharing the same MISA id are gathered into an instance of the class MISA, which itself contains one instance of the Chain_instance operator S ij [k], one instance of the Interface_oneside operator I Vor. ij [k] and one instance of the In-terface_string operator I ij [k] (cf. Fig 2). The classes implementing these three operators have the corresponding names (Chain_instances, Interface_oneside, Interface_strings).

8. 3 Figure S 9 :Figure S 10 :•

 3910 Figure S 9: Composition of the cMISA_constructor class

  8.2 Computing the MISA : script sbl-misa.py, module Multiple_interface_string_alignment.py Overview Multiple_interface_string_alignment.py creates the MISA from a specification file and the input data described above. First, it gathers the homologous structures together, by giving them the same MISA id, and further computes from this set of chains several operators (S ij [k], I Vor.

ij [k], P j,k

Acknowledgment. T. Dreyfus is acknowledged for insightful comments. This work has been supported by the French government, through the 3IA Côte d'Azur Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-19-P3IA-0002.

7 Supplemental: formal specification of the method

We formalize the notions introduced in the main text.

Voronoi interface restricted to a chain. We first process the Voronoi interface of a complex C i and collect all the interface amino-acids for a chain instance

Definition. 5 (Half interface) Consider the Voronoi interface model for the partners {A j } and {B j } in structure C i . The restriction of the Voronoi interface to chain S ij , or half interface for short, is the set of a.a. of this chain contributing at least one atom at the interface. This set is denoted I Vor.

ij

. The operator I Vor.

ij

[k] returns the one letter code of the k-th a.a. of S ij if this a.a. is at the interface, and ∅ otherwise.

Considering all chains associated with a given MISA id yields a set of a.a. at a given position, amidst which we seek the most frequent one:

The interface pool of chain A j at position k is defined as

where index i ranges over all complexes. The consensus a.a. at position k is the a.a. most frequent in the pool P j,k , with ties broken using the lexicographic on the one letter code of the amino-acids.

Interface strings. To encode properties of a particular chain instance S ij in a structure (bound or unbound), several parameters must be taken into account: the fact that this position is involved in interfaces or not, the potential absence of this position in selected crystal structures, and also the variability observed at that position (Def. 6). These pieces of information are precisely summarized in the so-called interface string, which is a string defined over the alphabet using the one letter code of a.a. plus the three symbols {- * }: Definition. 7 (Interface string) The interface string I ij of chain A j in structure C i or U i ) is the string with one character per amino-acid, defined as follows:

• (Case 1) k-th a.a. never at the interface i.e. P j,k = ∅: _ if this a.a. is absent from the crystal structure, and -otherwise. • (Case 2) k-th a.a. sometimes at the interface, but not in S ij i.e. P j,k = ∅ and I Vor. -* if a.a. is absent from the crystal; capital one letter code X for the consensus a.a., lowercase one letter code x otherwise.

Note that for a chain in an unbound structure, Case 3 is irrelevant since there is no interface.

Collecting all such strings for a given MISA id yields Def. 4.

Implementation details

In the following, we provide the implementation details, using in particular the notations of Section 7 for the operators.

Overview

sbl-misa.py is the main script of the package: it creates the MISAs, colors them and performs a structural comparison between the different chains of the same MISA. It uses different modules that we present here, through their main classes.

From its output, different complementary scripts can be used to further the analysis and/or parameterize the display of the MISAs. This section follows the organization of sbl-misa.py, and the different steps of the script refer to their numbering in Fig 7.

Record the colored MISA into an HTML file presenting simultaneously the four colorings

The cMISA_recorder creates the output files, by combining the Interface_string of each cMISA with the coloring legend provided by the color_engine of the cMISA. For each cMISA, the chains with the same output are grouped together to lighten the display. A .html file displaying simultaneously all the cMISA sharing the same MISA id is created in the odir/MISA directory. A sub-directory odir/MISA/single-coloring-figures is created, where each of the cMISA is stored individually, in both .pdf and .html format.

Coloring SSE

As explained in Section Colored MISAs, we color aa in interface strings to show changes in the H bonding network. To this end, we use DSSP [START_REF] Kabsch | Dictionary of protein secondary structure: pattern recognition of hydrogenbonded and geometrical features[END_REF] and https://swift.cmbi.umcn.nl/gv/dssp/. The corresponding color code reads as follows: The BSA is computed using the intervor "-bsa" option. For this coloring mode, only bound structures are displayed, as it would make no sense to do so for unbound structures, which necessarily have no BSA.

The ASA is computed using sbl-vorlume-pdb.exe, and the ∆_ASA is deduced from it, as explained in the main text. We display thus the ∆_ASA for the bound structures, and the ASA for the unbound structures. (If only bound structures were provided, the ∆_ASA won't be computable, and the sequence will be displayed in grey).

We use one color map for the BSA coloring, and two for the ∆_ASA coloring.

Coloring B-factor

The B-factor is extracted from the PDB file (and possibly computed from ANISOU, see main text, if only ANISOU is available). We use one color map for the B-factor coloring. Overview sbl-misa-diff.py allows to compare the interface of two homologous chains, by comparing the list of their residues at the interface.

Main steps

• Parse the spec_file

• Gather the data for the different interfaces, either in parsing an interface-file, or the MISA output.

• Compare the two interfaces 8.7 Structural comparison, script sbl-misa.py, module iRMSD.py

Overview sbl-misa.py finally also computes the i-RMSD for each pair of chains within each MISA, to compare the geometry of the different homologous chains.

Main steps and associated classes

Step 5. : Construction of the PDB interface files Main class : class Extract_PDB_interface Store the interface residues shared by each pair of chain into PDB files Creates for each pair of chain of each MISA two new .pdb files (one per chain), which only contains the interface residues shared by both chains. The selection of the residues at the interface is done thanks to the helper class ResSelector.

Step 6. : Computation of the i-RMSD for each pair of chain Main class : class Compute_interface_iRMSD_for_pairs Compute the iRMSD for each pair of PDB interface files Compute the iRMSD for each pair of chains by calling the script sbl-lrmsd-for-pdb-pair.exe for each pair of .pdb files precedently created. The l-RMSD is computed for three different 'grainmodes' (0: only the C-alpha, 1: the backbone, 2: all the atoms)

Step 7. : Analysis of the i-RMSD Main class : class Plot_tools Cluster the chains according to the iRMSD and record statistics on the iRMSD Once we have the i-RMSD for each MISA, we derive three figures from it :

• The first is a dendrogram showing the hierarchical clustering of the i-RMSD.

• The second is a distance matrix showing the value of the i-RMSD for each pair of chains.


Clément Moulin-Frier

Jean-Luc Schwartz

Julien Diard
email: julien.diard@upmf-grenoble.fr

Pierre Bessière
email: pierre.bessiere@imag.fr

Emergence du langage par jeux déictiques dans une société d'agents sensori-moteurs en interaction

Keywords: Evolutionnary Linguistic, Language Emergence, Bayesian Modeling, Cognitive Robotics, Multi-Agents Systems

In this paper, we show how some properties of human language could emerge from the primitive deixis function. For this aim, we model a society of sensori-motor agents able to produce vocalizations and to point to objects in their environnement. We show how principles of the Dispersion Theory [6] and the Quantal Theory [13] could emerge from the interaction between these agents.

Introduction

Depuis les années 70, des linguistes s'interrogent sur la possibilité de dériver le langage du non-langage [START_REF] Liljencrants | Numerical simulation of vowel quality systems : the role of perceptual contraste[END_REF]. Ils ouvrent ainsi la voie à des théories du langage "orientées-substance", qui mettent en valeur le rôle des contraintes morphologiques et cognitives dans l'apparition de formes universelles dans les langues du monde.

De cette évolution scientifique est né récemment un nouveau champ de recherche, parfois nommé Linguistique Evolutionnaire, qui cherche à montrer par la simulation informatique comment ces formes universelles peuvent émerger dans une société d'agents virtuels [START_REF] Steels | The synthetic modeling of language origins[END_REF][START_REF] Berrah | From form to formation of phonetic structures : An evolutionary computing perspective[END_REF]. C'est de cette problématique que traite cet article, en se basant sur des hypothèses fortes concernant l'apparition du langage chez l'Homme issues de certaines théories orientées-substance.

En effet, dans cet ensemble de théories, nous pouvons distinguer deux types d'approches. Le premier consiste à faire l'observation que le langage existe, puis à tenter d'expliquer ses tendances universelles par des contraintes périphériques au langage, provenant typiquement des propriétés des systèmes de perception et de production de la parole, sous l'hypothèse que les propriétés du langage émergent des conditions de communication. C'est le point de départ de théories telles que la théorie de la dispersion de Lindblom [START_REF] Liljencrants | Numerical simulation of vowel quality systems : the role of perceptual contraste[END_REF] ou la théorie quantique de Stevens [START_REF] Stevens | On the quantal nature of speech[END_REF]. Le deuxième type d'approches consiste quant à lui à tenter d'expliquer le langage lui-même par des fonctions primitives dont il dériverait, telles la mastication (théorie Frame/Content [START_REF] Macneilage | The Frame/Content theory of evolution of speech production[END_REF]) ou la déixis1 (théorie Vocalize to Localize [START_REF] Abry | Vocalize to localize ? a call for better crosstalk between auditory and visual communication systems researchers[END_REF]). Nous les nommerons respecti-vement "théories de la morphogénèse" et "théories de l'origine".

Notre hypothèse centrale est que les théories de la morphogénèse doivent elles-même dériver des théories de l'origine. Nous nous situons dans le cadre "Vocalize to Localize" [START_REF] Abry | Vocalize to localize ? a call for better crosstalk between auditory and visual communication systems researchers[END_REF] dans lequel la déixis a constitué un point de départ, un bootstrap à la naissance d'un langage articulé. Dans ce cadre, nous modélisons une société d'agents probabilistes (bayésiens) capables de montrer des objets dans un environnement. Nous montrons alors comment un code de parole respectant certains principes de la théorie de la dispersion et de la théorie quantique peut émerger de cette fonction de déixis.

Paradigme d'interaction

L'introduction d'une fonction de déixis dans nos agents est l'une des principales particularités de notre approche. En effet, nous pensons que le langage est né du besoin de "montrer de la voix". C'est la théorie "Vocalize to Localize" [START_REF] Abry | Vocalize to localize ? a call for better crosstalk between auditory and visual communication systems researchers[END_REF] selon laquelle la parole se serait constituée autour de trois composantes fondamentales : -Un système de production d'actions orofaciales pour émettre des vocalisations ; -Un système de perception phonétique pour structurer le flux sonore en unités perceptives intelligibles ; -Un système de pointage et d'attention partagée pour montrer le monde et produire du sens. C'est ce que nous modélisons en plongeant des agents capables de produire des gestes moteurs et de percevoir les percepts correspondants dans un environnement peuplé d'objets à nommer. Des agents sensorimoteurs évoluent donc dans un environnement dans lequel se trouvent des objets qu'ils peuvent identifier. Au cours du temps, ils se retrouvent de façon aléatoire deux par deux devant un objet O. Ils procèdent alors à ce que nous appelons un "jeu déictique", où un des deux agents prend le statut de locuteur, l'autre celui d'auditeur. Pour "montrer de la voix" cet objet, le locuteur propose, par un geste moteur M , une vocalisation. Par les lois de l'acoustique, ce geste moteur produit un percept sensoriel S perçu par l'auditeur (Figure 1). Les deux agents mettent alors à jour leurs connaissances en fonction de cette interaction (c'est-à-dire en fonction des valeurs de O et M pour le locuteur, et de O et S pour l'auditeur). Les jeux déictiques se succèdent au cours du temps, les agents se retrouvant aléatoirement en statut de locuteur ou d'auditeur.

Modélisation

Pour modéliser nos agents, nous utilisons le paradigme de Programmation Bayésienne des Robots (PBR) [START_REF] Lebeltel | Bayesian robot programming[END_REF] qui nous permet d'une part d'exprimer nos hypothèses et d'analyser nos résultats de façon mathématiquement claire grâce à des distributions de probabilité, d'autre part d'opérer sur les connaissances ainsi codées grâce à l'inférence bayésienne.

Fondements mathématiques

La Programmation Bayésienne des Robots vise à spécifier les comportements d'agents sensori-moteurs dans le cadre de la théorie bayésienne des probabilités. Un agent opère sur un ensemble de variables V , correspondant en général à des variables sensorielles (entrées de capteurs) et motrices (valeurs de commande). La méthode PBR se déroule alors en deux phases. La première est une phase déclarative dans laquelle on énonce des connaissances pertinentes au domaine. Elle a pour objectif le calcul de la distribution de probabilité conjointe P (V ). La seconde est une phase procédurale dans laquelle l'agent calcule des termes mathématiques d'intérêt en exploitant les connaissances énoncées préalablement. Plus précisément, il utilise la distribution conjointe pour répondre à des questions probabilistes de type P (X|Y ) (par exemple, connaissant la valeur de mes variables perceptives, quelle est la distribution de probabilité sur mes variables motrices ?). Dans le cadre de cette article, nous ne pouvons détailler plus précisément notre cadre computationnel, le lecteur intéressé se reportera à [START_REF] Lebeltel | Bayesian robot programming[END_REF] pour les aspects généraux de la PBR et à [START_REF] Moulin-Frier | Jeux déictiques dans une société d'agents sensori-moteurs[END_REF] pour un exposé plus détaillé de nos travaux.

Modélisation des agents

Dans le cadre du paradigme d'interaction exposé Figure 1, nous retenons quatre variables par agent : -O L : l'ensemble des objets devant lesquels l'agent s'est retrouvé en situation de locuteur, -M : l'ensemble des gestes moteurs qu'un agent est capable de produire en situation de locuteur, -S : l'ensemble des percepts sensoriels qu'un agent peut percevoir en situation d'auditeur, -O A : l'ensemble des objets devant lesquels l'agent s'est retrouvé en situation d'auditeur. Pour chacun des agents, l'objectif est alors de calculer la distribution conjointe sur ses quatre variables :

P (O L ∧ M ∧ S ∧ O A ).
P (O L ∧ M ∧ S ∧ O A ) = P (O L ).P (M |O L ).P (S|M ∧ O L ).P (O A |S ∧ M ∧ O L )
.

(

) 1
puis on la simplifie grâce aux hypothèses ci-dessus. Ainsi :

-

P (O L ∧ M ∧ S ∧ O A ) ∝ P (M |O L ).P (S|M ). P (S|O A ) O A P (S|O A ) . (3)
Par cette équation, chaque agent est capable de calculer la distribution de probabilité conjointe sur l'ensemble de ses variables d'intérêt (O L , M , S et O A ). Ainsi, il est en mesure de calculer n'importe quelle distribution conditionnelle. Remarquons que comme les distributions P (M |O L ) et P (S|O A ) sont apprises au cours des jeux déictiques par chacun des agents, la distribution conjointe de l'équation (3) évolue au cours du temps. La section suivante montre comment cette évolution peut mener à la création d'un code de parole commun en accord avec certains principes des théories de la morphogénèse.

Simulations et analyses

Le comportement communicatif

Le principe de ce comportement est de tirer un geste moteur qui maximise la probabilité d'être compris par l'auditeur. Ainsi, devant un objet o i , l'agent locuteur sélectionne un geste M de façon à maximiser la probabilité P (O A = o i |M ). Or, par inférence bayésienne sur l'équation (3) et d'après la définition de P (S|M ), nous avons

P (O A = o i |M ) = P (O A = o i |S = percept(M )) .
(4) Ce comportement va donc chercher à satisfaire le modèle locuteur, en tirant des gestes M dont le percept correspondant permettra à l'auditeur d'inférer facilement l'objet o i par la question probabiliste P (O A |S). On observe alors l'émergence d'un code de parole commun (Figure 3). Fig. 3: Comportement communicatif : état de la simulation après 100 000 jeux déictiques. On observe que les agents ont organisé les gestes moteurs produits pour chaque objet de façon satisfaisante (taux de compréhension autour de 80%).

Ce comportement peut être analysé en regard de deux théories de la morphogénèse : la théorie de la dispersion [START_REF] Liljencrants | Numerical simulation of vowel quality systems : the role of perceptual contraste[END_REF] et la théorie quantique [START_REF] Stevens | On the quantal nature of speech[END_REF]. La première stipule que les systèmes phonologiques des langues du monde (en particulier les systèmes de voyelles) tentent de maximiser les distances inter-phonèmes. Nous voyons Figure 3 que cet effet émerge de nos simulations (les P (M |O L ) se sont dispersés, donc les P (S|O A ) également car ici la fonction percept est la fonction identité). La théorie quantique quant à elle stipule que les langues du mondes exploitent les non-linéarités présentes dans la transformation articulatori-acoustique en choisissant des phonèmes dans des zones stables, où des variations sur les paramètres moteurs ont peu de conséquences sur les paramètres perceptifs et en utilisant la zone de non-linéarité comme frontière entre catégories phonétiques. Or, si nous introduisons une non-linéarité dans la fonction percept en la définissant comme une sigmoïde, nous observons que nos agents choisissent des commandes motrices de part et d'autre de celle-ci (Figure 4).

Le comportement hybride

P (M |O L = o i ∧ O A = o i ) = P (M |O L = o i ).P (O A = o i |S = percept(M ))
.

(

Conclusions et perspectives

Le cadre de simulation présenté ici montre comment on peut faire émerger des principes de mise en forme (tels que la théorie de la dispersion ou la théorie quantique) d'un principe de dérivation plus général, faisant dans notre cas un lien entre l'émergence du langage et un comportement plus primitif de déixis. Ce comportement est attesté et bien décrit chez les primates non humains [START_REF] Leavens | Deriving reference : bipedalism, encephalization, and the referential problem space[END_REF][START_REF] Slocombe | Functionally referential communication in a chimpanzee[END_REF].

Nous avons commencé à implémenter ces principes sur un modèle réaliste de conduit vocal et de système auditif, les simulations conduisent à une dispersion au sein du triangle vocalique, conformément aux résultats classiques des travaux de Lindblom et suivants [START_REF] Liljencrants | Numerical simulation of vowel quality systems : the role of perceptual contraste[END_REF]. Pour passer de vocalisations simples comme les voyelles à des vocalisations plus complexes comme les syllabes, nous nous appuierons sur un autre comportement précurseur, celui des cycles mandibulaires associés à la mastication, et produisant, selon la théorie Frame/Content, un principe de modulation important pour passer à des séquences complexes [START_REF] Macneilage | The Frame/Content theory of evolution of speech production[END_REF].

Dans cette démarche, nous privilégions systématiquement la recherche de comportements précurseurs et de contraintes émergeant de ces comportements [START_REF] Schwartz | Eléments pour une morphogenèse des unités du langage[END_REF]. Nous nous distinguons par là d'autres travaux cherchant à faire émerger la complexité à partir d'un nombre d'hypothèses minimum [START_REF] Oudeyer | Aux sources du langage : l'autoorganisation de la parole[END_REF]. Notre position est au contraire que les formes sont contraintes par les propriétés des systèmes en interface et que c'est en intégrant le plus grand nombre possible de connaissances sur ces systèmes (production, perception de vocalisations) que les simulations s'approcheront le plus des réalités linguistiques.

Fig. 1 :

 1 Fig. 1: Un jeu déictique entre deux agents

  Pour cela, nous faisons les hypothèses suivantes : -H 1 : la présence des objets dans l'environnement est uniforme (tant en statut de locuteur que d'au-diteur). -H 2 : les agents sont capables d'estimer correctement le percept correspondant à un geste donné, ils connaissent donc la distribution P (S|M ). Il s'agit d'un modèle direct de copie d'efférence (estimation des paramètres sensoriels à partir des commandes motrices) classique en Sciences Cognitives [3]. -H 3 : les agents apprennent au cours des jeux déictiques la distribution P (M |O L ), c'est-à-dire la distribution des gestes moteurs pour chaque objet devant lequel ils se sont retrouvés en situation de locuteur. -H 4 : de même, les agents apprennent au cours des jeux déictiques la distribution P (S|O A ), c'est-à-dire la distribution des percepts entendus pour chaque objet devant lequel il se sont retrouvés en situation d'auditeur. Pour calculer la distribution conjointe, on la décompose en un produit de termes plus simples à spécifier grâce au théorème de Bayes

  Dans cette section, nous exposons trois simulations correspondant à trois comportements différents pour nos agents. Nous ne nous attachons pas pour l'instant à la modélisation d'un conduit vocal et d'une oreille réaliste. Les variables M et S sont définies sur un espace unidimensionnel borné et la fonction percept définissant la transformation articulatori-acoustique est la fonction identité (on a donc S = M ). Les trois comportements diffèrent alors par la question probabiliste que chaque agent pose à sa distribution conjointe (équation (3)) pour tirer un geste moteur M en situation de locuteur dans un jeu déictique.

4. 1 .

 1 Le comportement réflexeCe premier comportement est un comportement naïf qui permet de montrer que la seule prise en compte des intérêts du locuteur ne permet pas de faire émerger un code de parole commun dans la société d'agents. Devant un objet o i , il consiste simplement en un tirage du geste moteur M selon la distribution P (M |O L = o i ) (modèle locuteur). Ce comportement favorise donc les gestes déjà souvent tirés devant un objet donné sans se soucier du besoin de l'auditeur. Il ne peut donc mener à l'émergence d'un code de parole commun, comme nous l'observons Figure2.

Fig. 2 :

 2 Fig.2: Comportement réflexe : état de la simulation après 100 000 jeux déictiques dans une société de quatre agents et un environnement de quatre objets. Les quatre fenêtres supérieures correspondent à l'état des quatre agents. Dans chaque fenêtre, les quatre courbes représentent les distributions P (M |O L ) pour chacun des objets de l'environnement. Nous constatons ici qu'il n'y a pas de cohérence dans les gestes produits par chaque agent pour chacun des objets. La fenêtre inférieure représente le taux de compréhension dans la société au cours du temps (avec un facteur 100, 1000=100 000 jeux déictiques), c'est-àdire le pourcentage de jeux déictiques dans lesquels le percept produit par l'agent locuteur a pu permettre à l'agent auditeur d'inférer le bon objet par la question probabiliste P (O A |S) (sur les 100 derniers jeux). On observe qu'il reste autour de 25%, ce qui correspond au taux du hasard (il y a quatre objets).

  Alors que le comportement réflexe cherche seulement à satisfaire le modèle locuteur P (M |O L ) et le comportement communicatif seulement le modèle auditeur P (O A |S), ce comportement hybride vise à satisfaire

Fig. 4 :

 4 Fig. 4: a) et b) : deux fonctions percept avec une nonlinéarité (M en abscisse, S en ordonnée). c) et d) : état de la simulation dans une société de quatre agents et un environnement de deux objets après 100 000 jeux déictiques pour chacune des fonctions percept a) et b), respectivement. On observe que les agents choisissent des gestes moteurs de part et d'autre de la non-linéarité, et que le taux de compréhension atteint 100%.

  ) Cette distribution est donc le produit de deux termes, le premier P (M |O L = o i ) correspondant au comportement réflexe, le deuxième P (O A = o i |S = percept(M )) au comportement communicatif. Ce comportement mène à l'émergence d'un code de parole commun dans la société d'agents (avec un taux de compréhension de 100%) et possède les même propriétés que le comportement hybride : il réalise de la dispersion et exploite les non-linéarités présentes dans la fonction percept. Il nous semble cependant plus intéressant que le comportement communicatif car il introduit une contrainte articulatoire grâce au terme P (M |O L = o i ) et fournit une base mathématique à la théorie, ni perceptive ni motrice, mais perceptuomotrice que nous défendons pour la communication parlée : la Théorie de la Perception pour le Contrôle de l'Action[START_REF] Schwartz | Eléments pour une morphogenèse des unités du langage[END_REF].

  H 1 : le terme P (O L ) est constant (il n'interviendra donc pas dans les calculs). -H 3 : le terme P (M |O L ) est appris par les agents au cours des jeux déictiques. Nous le considérerons comme une famille de distributions gaussiennes (une par valeur de O où le terme P (S|O A ) est appris par les agents au cours des jeux déictiques. Nous le considérerons comme une famille de distributions gaussiennes (une par valeur de O A ), dont les paramètres à apprendre par l'agent sont les moyennes µ O A et les variances V O A . Nous appellerons cette distribution le modèle auditeur. Nous obtenons alors l'expression simplifiée :

	P (S|M ) =	1 si S = percept(M ), 0 sinon.	. Nous appelle-
	rons cette distribution le modèle "copie d'efféren-
	ce". -H O A	P (S|O A )	.	(2)

L ), dont les paramètres à apprendre par l'agent sont les moyennes µ O L et les variances V O L . Nous appellerons cette distribution le modèle locuteur. -H 2 : le terme P (S|M ∧ O L ) se simplifie en P (S|M ) et est connu par les agents. Nous considérons qu'il existe une fonction percept telle que S = percept(M ). La distribution P (S|M ) s'exprime alors comme une distribution Dirac, c'est-à-dire 4 : la distribution P (O A |S ∧ M ∧ O L ) se simplifie en P (O A |S). Par application des règles de Bayes, et comme P (O A ) est uniforme (hypothèse H 1 ), on a P (O A |S) = P (S|O A )

Déixis, du grec deiktikos, signifie "action de montrer"