Pavel Arapov
email: arapov@i3s.unice.fr

Michel Buffa
email: buffa@i3s.unice.fr

Amel Ben Othmane
email: amel.ben-othmane@inria.fr

WikiNEXT: a wiki for exploiting the web of data

Keywords: K.4.3 [Organizational Impacts]: Computer-supported collaborative work Algorithms, Performance, Design, Experimentation, Languages Semantic Web, Web2.0, Wikis, Semantic Wikis, Knowledge Management, Web Applications

This paper presents WikiNEXT, a semantic application wiki. WikiNEXT lies on the border between application wikis and modern web based IDEs (Integrated Development Environments) like jsbin.com, jsfiddle.net, cloud9ide.com, etc. It has been initially created for writing documents that integrate data from external data sources of the web of data, such as DBPedia.org or FreeBase.com, or for writing interactive tutorials (e.g. an HTML5 tutorial, a semantic web programming tutorial) that mix text and interactive examples in the same page. The system combines some powerful aspects from (i) wikis, such as ease of use, collaboration and openness, (ii) semantic web/wikis such as making information processable by machines and (iii) web-based IDEs such as instant development and code testing in a web browser. WikiNEXT is for writing documents/pages as well as for writing web applications that manipulate semantic data, either locally or coming from the web of data. These applications can be created, edited or cloned in the browser and can be used for integrating data visualizations in wiki pages, for annotating content with metadata, or for any kind of processing. WikiNEXT is particularly suited for teaching web technologies or for writing documents that integrate data from the web of data.

Introduction / motivation

Since Ward Cunningham's wiki in 1995, a lot of wiki engines have appeared, and propose common functionalities that we can see as the "DNA of wikis": they allow users to easily create, link together, update, and delete wiki pages in a collaborative way. Rapidly some wiki engines provided "industrial" features adapted to private organizations [START_REF] Buffa | Intranet Wikis[END_REF] like backend integration, security management, versioning, possibility to embed HTML forms, templates, write small applications within the wiki pages, plugins and extensions systems, attachments to wiki pages, etc. We call this trend "application wikis" 1 where we find TWiki, FOSWiki, TikiWiki, Confluence, XWiki, Mindtouch Core (previously DekiWiki), etc. Another interesting trend appeared in 2005 with "semantic wikis": wikis that use the wiki way for creating/maintaining ontologies (data models) collaboratively and/or for annotating the content of the wiki using ontologies, and enable reasoning on the data. See [START_REF] Buffa | SweetWiki: A semantic wiki[END_REF] [START_REF] Krotzsch | Reasoning in semantic wikis[END_REF] for an overview. Our group developed two semantic wikis: SweetWiki [START_REF] Buffa | SweetWiki: A semantic wiki[END_REF] in 2006-2010 and SweetDeki [START_REF] Gandon | ISICIL: Information Semantic Integration through Communities of Intelligence online[END_REF] recently, giving us a strong experience in that field. Furthermore, we authored the online HTML5 certification course for the W3C 2 and had to develop many HTML5 interactive examples in HTML/JavaScript/CSS, using the jsbin.com web-based IDE. The integration of these examples in the moodle 3 e-learning platform (that the W3C imposes to the authors) was painful and very limited (using just links or iframes): we imagined rapidly how it could have been done using a wiki. Since 2012, we are also in charge of DBPedia.fr, the French version of DBPedia.org, a semantically structured version of Wikipedia that accepts SPARQL 4 requests via REST web services. For this project, we started to develop WikiNEXT, a wiki engine that could act as a front-end for DBPedia: one could write applications from within the wiki, which can talk to DBPedia.fr, could write tutorials made both of text and application code that can be modified collaboratively. More generally we designed WikiNEXT for integrating HTML5/JavaScript applications that can take advantage of the web of data (in other words, process data from different data sources, eventually cache them, display and manage them in applications developed in the wiki). Writing semantic web applications can be cumbersome as there are lots of technologies to handle: RDF, RDF/S, SPARQL, datasets use different models/vocabularies/ontologies that need to be learnt, etc. WikiNEXT seamlessly integrates the editing of "documents" and the editing of "applications within the documents", hopefully with some templates, examples, tutorials, possibility to add and use external JavaScript libraries, etc. Finally, we will see that we considered each wiki page as "an application" by itself, not as a document with some restricted rectangular areas with interactive applications inside. Indeed, some applications can be "not visible": for example, applications that collect data from DBpedia.org for enriching the text of the document (highlight words that are Wikipedia entries, add semantic annotations, etc.), whereas others can be complex data visualizations with interactive parameters. This paper is organized as follows: in Section 2, we give an overview of wikis, focusing on application and semantic wikis and we explain how they are used as tools to address specific knowledge needs. We also present the concept of web-based IDEs that became popular these last years and influenced WikiNEXT a lot. In section 3 we will present WikiNEXT design principles, its architecture and its implementation. In section 4 we describe how users could use our wiki to develop semantic web applications and detail one example. Section 5 explains how we tested and validated WikiNEXT on a test group of web developers, and how we will conduct more tests in a near future. Finally, Section 6 concludes and presents some perspectives.

Convergence between wikis, application wikis, semantic wikis and web-based IDEs 2.1 Application wikis

Among successful Web 2.0 platforms, wikis, exemplified by Wikipedia, are known for promoting the convergence of collaborative writing. Besides enabling people to contribute content to the web, wiki usages have also been extended to other domains. Some wikis, called "applications wikis", rapidly integrated features for "programming in the pages". TWiki for example, in 1998, pioneered this approach by proposing sorts of macros for generating tables, forms, and integrated a system of plugins for extending the standard set of macros. Rapidly other wikis such as Jot (that became Google Sites), XWiki or Mindtouch 7 , integrated one or several script languages for integrating "dynamic parts" in the pages. This let developers collectively build web applications within wikis including data modeling, data management, data processing and data presentation. Editing, copying and pasting code from other pages: Ward Cunningham, the author of the very first wiki, calls it "the wiki way" [START_REF] Leuf | The Wiki way: quick collaboration on the Web[END_REF] and turns the wiki into an easy tool for "learning from the others' work". The source scripts and data of such wiki applications are invisible to web users who read pages, but editing a page reveals the code behind.

Here are some examples of macros/applications written using such scripts: instantiate a whole application (a spreadsheet, a mini blog, a photo gallery…), insert a table of contents, a form, etc. These macros are either pre-defined or written by users using a scripting language (Velocity8 or Groovy 9 for XWiki, DekiScript, -a derivative of JavaScript-for Mindtouch Core, etc.). The set of macros may use server-side plugins/extensions. In this case the developer of plugins/extensions needs to have access to the source code of the wiki, or at least to an SDK (in Java for XWiki, perl for TWiki/FOSWiki, PHP or C#/Mono for Mindtouch), and sometimes needs some external tools (like the full Mono development environment in the case of Mindtouch). Most of these application wikis generate business and have been in development for several years, a large set of plugins/extensions is available. However, they are heavyweight tools and only a few of them integrate emergent technologies such as WebSockets10 or HTML5 advanced APIs, as they target private organizations where browsers are generally several generations late.

Semantic wikis

The semantic web is about adding formal semantics (metadata, knowledge) to the web content for the purpose of more efficient access and management. It enables structural and semantic definitions of documents providing completely new possibilities: intelligent search instead of keyword matching, query answering instead of information retrieval, suggestions, collaborative filtering, etc. Since its vitality depends on the presence of a critical mass of metadata, the acquisition of this metadata is a major challenge for the semantic web community. However, just as the success of the World Wide Web can be attributed to the ease of use and ubiquity of web browsers, we believe that the unfolding of the semantic web vision depends on users getting powerful but easy-to-use tools for managing their information. Unlike HTML, which can be easily edited in any text editor, the W3C RDF language, the most popular standard for representing knowledge on the web of data, is more complicated to author and does not have an obvious presentation mechanism [START_REF] Kobilarov | Media meets semantic web-how the bbc uses dbpedia and linked data to make connections[END_REF].

As part of this process of simplification, researchers investigated different approaches for producing and authoring knowledge. Semantic wikis is one of them. Semantic wikis such as Semantic Media Wiki [START_REF] Krötzsch | Semantic mediawiki[END_REF], IkeWiki [START_REF] Schaffert | IkeWiki: A semantic wiki for collaborative knowledge management[END_REF], SweetWiki [START_REF] Buffa | SweetWiki: A semantic wiki[END_REF], started as research projects. They allow users to add semantic context to wiki pages content while preserving the simplicity and the collaborative essence of wikis. Several semantic wiki engines also integrate features of application wikis. A semantic wiki enables users to additionally describe resources in a formal language: RDF (Resource Description Framework), OWL (Web Ontology Language), or Conceptual Graphs [START_REF] Oren | How semantics make better wikis[END_REF]. Annotations added to wiki pages are stored in a knowledge base. These annotations are then used to augment a wiki with functionalities like enhanced navigation, search and retrieval or reasoning. SPARQL is the standard W3C language 11 for querying RDF data. The RDF data-model is a directed, labeled, multi-graph and, thus, SPARQL is essentially a graph-matching query language. Annotations in semantic wikis are formal and possibly semantic, e.g. they are formally defined, and possibly use ontological terms. Systems such as Semantic MediaWiki [START_REF] Krötzsch | Semantic mediawiki[END_REF], based on the Media Wiki engine, or SemperWiki [START_REF] Oren | SemperWiki: a semantic personal Wiki[END_REF] require to use special wiki syntax or to directly embed RDF in order to add semantic annotations to wiki pages. While this is an open approach in the spirit of wiki principles, this can lead to semantic heterogeneity problems since any user can use its own vocabulary to add annotations in a document, making them difficult to re-use. A system like IkeWiki [START_REF] Schaffert | IkeWiki: A semantic wiki for collaborative knowledge management[END_REF] combines plain-text feature of wikis and a dedicated triples-based form interface to help users annotating content by re-using existing ontologies, while OntoWiki [START_REF] Auer | OntoWiki-A tool for social, semantic collaboration[END_REF] can be used as a complete ontology editor, with a user-friendly interface that offers different views and browsing and editing interfaces over existing data.

Eventually, it seems important to reference DBPedia [START_REF] Auer | Dbpedia: A nucleus for a web of open data[END_REF], a project that aims to represent Wikipedia content in RDF, as well as other semantic wikis, like SweetWiki [START_REF] Buffa | SweetWiki: A semantic wiki[END_REF] which does not focus on ontology population but on using semantic web technologies to let users tag their pages and organize those tags, focusing on pages meta-data rather than modeling content of those pages.

Today, as detailed in [START_REF] Meilendera | Semantic wiki engines: a state of the art[END_REF], semantic wikis which are still active have chosen different approaches: (1) extend their wiki with different functionalities or (2) externalize some parts . In the first category we find, for example, the Halo12 extension of Semantic Media Wiki which proposes forms, auto-completion, a WYSIWYG editor, the integration of audios and videos files, and the integration of a SPARQL endpoint, or more complex extensions such as MoKi [START_REF] Rospocher | Lindstaedt: MoKi: the Modelling wiKi[END_REF] (focuses on enterprise modeling) and OWiki [START_REF] Di Iorio | Ontology-driven generation of wiki content and interfaces[END_REF] (ontology-driven generation of wiki content), also based on Semantic Media Wiki. In the second category we find the KiWi project (Knowledge in Wiki [START_REF] Schaffert | Kiwi -knowledge in a wiki[END_REF]), which involved authors of IkeWiki, or SweetDeki [START_REF] Buffa | ISICIL: Semantics and Social Networks for Business Intelligence[END_REF] the wiki that succeeded to SweetWiki [START_REF] Buffa | SweetWiki: A semantic wiki[END_REF], and has been integrated in the ISICIL ANR project [START_REF] Buffa | ISICIL: Semantics and Social Networks for Business Intelligence[END_REF]. These projects tried to put semantic wikis in an industrial context, including the issue of project management.

There are other issues, as illustrated by the introduction text to the first workshop about Semantic Web programming 13 (2012): "The Web of Data is growing at an enormous pace. However, the development of dedicated software applications, capable to deal efficiently in information-rich spaces, of which the Semantic Web is one dimension, is not yet mainstream. Reasons for that include one (or more) of the following research issues: lack of integrated development environments (IDEs, such as Visual Studio and Eclipse), poor programming language support, lack of standard test beds and/or benchmarks, inadequate training, and perhaps the need for curricula revision. Properly addressing these issues requires interdisciplinary skills and the collaboration between academia and industry". This comment is true for many semantic wiki engines too. For example, while a lot of plugins/extensions are available for Semantic Media Wiki, adding a new functionality or plugging a new external data sources often requires server side programming and modifying several files. Modern webbased IDEs are presented in the next section. They influenced WikiNEXT design principles and are a possible solution to this inconvenience.

Web based IDEs

Writing applications using a web based IDE is not new, we saw early experiments in the 90's, such as WebWriter [START_REF] Crespo | WebWriter: A browserbased editor for constructing web applications[END_REF], advanced tools based on interpreted languages and Ajax appeared as soon as this technology was born [START_REF] Serrano | Hop: a language for programming the web 2. 0[END_REF]. With Ajax and the development of JavaScript frameworks, online source code editors kept improving. Editors like Code Mirror 14 , ACE Cloud9 15 or ternjs 16 propose features such as syntax highlighting, autocompletion, and have been used by many online IDEs 17 . Wellknown multi-language IDEs such as compile-online.com, ideone.com or compilr.com are useful for showing to students small examples of code written in a variety of programming languages (Java, C, C++, Lips, JavaScript, etc.), the code is sent to a server, compiled, executed, and output results are displayed in another iframe in the web interface. Now, some lightweight online IDEs are well known by web developers: jsbin.com, jsfiddle.net or tinkerbin.com integrate popular libraries like jQuery, Sencha, Dojo, etc. and propose really interesting features such as real time editing of HTML/CSS/JavaScript code, instant preview when files are updated, versioning using problems cited. 14 http://codemirror.net/ 15 http://ace.c9.io/ 16 http://ternjs.net/ 17 See http://en.wikipedia.org/wiki/Comparison_of_JavaScript-based_source_code_editors

Git 18 , making all examples developed online sharable as each version has a unique URL. Most of these tools however, are sandboxes for trying code online or for writing examples that can be shared, like the ones we developed using jsbin.com and integrated in the HTML5 course we wrote for the W3C. Finally, there are more ambitious online tools 19 for web development in the cloud, like cloud9ide.com, shiftedit.net, coderun.com, codeanywhere.net, that makes possible creating real multi files web projects with the possibility to distinguish client-side and server-side code, with a complete team of programmers working on them. WikiNEXT can be considered as a powerful web-based IDE as it proposes most of the features found in the last generation of these tools (instant preview, run code in the browser, etc.).

WikiNEXT

Application wikis, semantic wikis, modern online IDEs: we tried to mix them all with WikiNEXT; we considered each page as a web application, composed of HTML/CSS/JavaScript code. We provided several editors in the browser, for editing the text and the JavaScript source code, and provided an API for "talking with the wiki engine internals". Consider the text content of a wiki page as data: the application part of the page (JavaScript code) can collect semantic data from an external source, use them for annotating the document, extract information from the text for suggesting tags, add graphic visualizations, save data in a cache, etc. WikiNEXT has been designed for making the process of writing such application easier, and without the need to use any extra tool. ScraperWiki20 is a wiki that pioneered this kind of external data integration. It has been designed for monitoring government data over time (to see how public money is used). Its users upload or create/update Python/Ruby/PHP code for data scrapers directly in the wiki editor that will collect, visualize and save government data from external sources (mainly URL of web pages) [START_REF] Kienle | Open Data: Reverse Engineering and Maintenance Perspective[END_REF]. WikiNEXT proposes to go further with a real client side application layer that enables richer data manipulation using semantic web standards, an API for caching and updating data, and more.

WikiNEXT [START_REF] Arapov | WikiNext, a JavaScript semantic wiki[END_REF] is both (1) a semantic wiki (e.g. a wiki that uses technologies from the semantic Web to embed formalized content, structures and links, that can reason on this knowledge, etc.) and (2) an application wiki as it provides an API and a platform of services to use from applications written in the wiki and (3) a modern web-based IDE. Unlike a traditional wiki, pages in WikiNEXT are considered as applications that can be managed in the wiki way: look at an application, clone it, look at the code, and modify it. The application layer of a wiki page is made of JavaScript code that can be edited the same way as the text content of the page, but using a dedicated, embedded source code editor. It is then possible from the JavaScript code to manipulate the Document Object Model of the page and to perform interactive actions. Any HTML based GUI element can be added in a page with JavaScript code for handing events. Talking with the wiki internals or requesting external data sources is done through the WikiNEXT API. WikiNEXT provides tools for plugging external JavaScript libraries such as D3.js 21 .

The prototype is open source 22 and available online 23 . It includes many examples and tutorials, e.g. for writing applications that exploit the semantic database DBPedia.org, and use the WikiNEXT API, for processing the semantic data collected and enhancing wiki documents. This makes WikiNEXT a good tool for teaching semantic web technologies, without the need to install any software. It is also a good tool for collecting data and integrating them in documents.

The next section gives an overview of the API architecture and how WikiNEXT has taken advantages of recent approaches and trends in web development.

Architecture overview

In order to realize the self-organized reuse in software engineering -in particular when performed under the agile software development paradigm [START_REF] Rech | Using wikis to tackle reuse in software projects[END_REF] -wikis can be seen as a lightweight platform for exchanging reusable artifacts between and within software projects. On the other hand, based on the latest development in web technologies, improved JavaScript engines, and the introduction of HTML5/JavaScript, there are now several efficient browser-based source code editors.

Combined with the existing functions of wiki, e.g. collaboration and documentation, WikiNEXT is expected to support software communities to develop JavaScript applications in a fully online manner which makes it a mixture between a wiki and a web based IDE. An article/page in WikiNEXT is considered as a web application: users can use different editors (1) a HTML or (2) a JavaScript editor (see Figure 1).

The WikiNEXT JavaScript API provides numerous objects, events and methods that can be used by applications developed within the wiki. In an MVC paradigm, the JavaScript part of the page will be the business layer and the Controller, the View part will be done in the HTML/CSS of the page, and the Model part is composed of RDF metadata (see Figure 2). WikiNEXT articles/pages are associated with a set of metadata that describe their basic characteristics: Title, Article, Author, Contributor, Date, etc. But articles are also containers: they can hold metadata created manually or added automatically by an application (see Figure 3). WikiNEXT uses schema.org 24vocabularies to describe the default structure of articles, users' details, etc. Each article is represented semantically as a named graph made of RDFa/RDFalite annotations, whose name is based on its URI (Uniform Resource Identifiers). We call this graph the Local Knowledge Base. The full stack of wiki articles give us the Global Knowledge Base, the union of all the named graphs.

The metadata generated by applications embedded in the page/article, e.g. a list of countries resulting from a request or a web services call to DBPedia.org, are also represented in the page as RDFa/RDFaLite25 annotations. WikiNEXT parses these annotations and saves them in the integrated triple store [START_REF] Hernández | A JavaScript RDF store and application library for linked data client applications[END_REF].

We use a modified version of RDFStore-js [START_REF] Hernández | A JavaScript RDF store and application library for linked data client applications[END_REF], which supports SPARQL 1.1. The traditional page content is stored as objects in a MongoDB26 database, and indexes for the textual page content are generated this way.

Implementation

WikiNEXT has been written from scratch in JavaScript and relies on the Node.js HTTP server and on the MongoDB database. Node.js27 (Node) is an I/O environment built on top of Google Chrome's JavaScript runtime -essentially, a serverside implementation of a JavaScript interpreter. MongoDB is a NoSQL document-oriented database. Instead of storing serialized objects in rows and columns as one would with a relational database, it stores objects using a binary form of JSON objects, called BSON28 . RDFstore-js triplets are also persisted in MongoDB, then, RDF triplets are both accessible through RDFstore-js using SPARQL, but also directly through MongoDB, Notice that some researchers have also tried to use MongoDB directly as a triple store [START_REF] Tomaszuk | Document-oriented triple store based on RDF/JSON[END_REF].

WikiNEXT also takes advantage of templates for creating pages with semantic annotations. Our wiki uses the Mustache29 template engine that has a "logic-less" syntax: it doesn't rely on procedural statements (if, else, for, etc.).

With these technological choices, we tried to minimize the number of languages and the number of data formats used in WikiNEXT: JavaScript can be used in the whole process of development both for programming and for persistence, both on the client side and on the server side.

In the next section, we detail a typical example of an application that can be developed in this environment.

Scenario

In this section, we describe a concrete scenario of usage of WikiNEXT: we write an application in a wiki page that uses the DBPEdia.org SPARQL endpoint to retrieve linked data about cities (description, pictures, population, etc.). This application requests DBPedia.org, gets the results and uses them to populate wiki pages with semantic annotations using RDFaLite. These annotations are persisted in the wiki RDF store and we will show how this data can be reused by other applications, e.g. for a semantic mashup that displays all collected metadata about cities on a single map page.

City application

This application is available on WikiNEXT 30 and can be run, edited or cloned by any user. It is composed of:

1. A View form: as shown in Figure 4, it allows users to look for information about a city and creates a new page with the found information or makes an update if the page for this city already exists in the wiki.

2. An application code that finds information about a city by querying DBPedia.org 31 's SPARQL endpoint.

3. A template for displaying the data collected, we use a template that gives a nice layout.

4. Some code for creating a new page: in order to reuse information, WikiNEXT offers the possibility to create pages that contain both information about cities and semantic annotations.

30 http://wikinext.gexsoft.com/wiki/519e04c580194c4178000001 31 http://dbpedia.org/

Retrieving data

The WikiNEXT API proposes some "default" rendering for displaying RDF data, we call this mode "sparqlOnFly", see the example of Figure 6. On the other hand, it is possible to create some "template pages" in order to have a more meaningful and structured view of the information, as shown in Figure 7. The HTML of the corresponding template page is shown in Figure 10. The principle is simple to use and permits the decoupling of the UI (User Interface) definition from the data. The user needs first to define the UI in the HTML part as the example below:

Welcome to {{city}}.

In the JavaScript part data will take this form:

var data = {city: 'Nice'};

The result will be:

Welcome to Nice.

WikiNEXT proposes an API with various features to cache the data collected from external data sources, refresh existing ones, etc. Figure 5 shows the complete functional diagram of the WikiNEXT architecture.

Processing data

The template shown in Figure 10 both handles the display of the data retrieved from DBPedia, and the RDFa Lite annotations.. For example, a city is defined by: <span about={{_id}} typeof="City"> <h1 property="name">{{city}}</h1> … </span>

In the code above we defined an object with identification "{{_id}}" and that has an RDF type "City", from the vocabulary http://schema.org/City. This vocabulary states that a City has a property "name", and we will give it the value "{{city}}". This syntax comes from the templating engine, see above.

The JavaScript code below fills the template with real values:

Reusing data

To demonstrate data reuse, we try to recombine data sources on the wiki, creating what is typically known as mash-up by developing a web application that combines all the created cities on WikiNEXT. The Figure 8 displays the output map of the wiki application that retrieves the location of all cities available in our knowledge base and shows all cities on a map and in a table.

WikiNEXT uses SPARQL to query its knowledge base (see Figure 9). Hence, queries can be written without having to deal with semantic heterogeneity problems since we use predefined ontologies.

Besides the individual display of data, WikiNEXT enables semantic search. Figure 11 shows the search result of a keyword "Paris" for example. Since we have created a web page that contains an article named Paris, the semantic search algorithm gives us as result that Paris is both the name of a web page (article) and a city in WikiNEXT.

Testing and validating WikiNEXT

We wrote some online tutorials, using WikiNEXT itself, and set up a testing protocol32 . We choose a group of people with a diverse level of web programming knowledge, and asked them to complete various tutorials available in WikiNEXT, that go from writing very simple examples to developing the application described in section 5. Our testers were all members of the Wimmics33 research group at the INRIA/I3S Research Laboratory (master students, PhD students, engineers, researchers, visitors). We measured the time spent for completing each tutorial, observed behaviors of our testers and took note of all the problems encountered. Finally we conducted an interview, and asked them to compare this experience with classical web development techniques for performing the same tasks (e.g. using Java/Eclipse). Results34 showed that the system is easy to use, saves time and enables easy data reuse between applications, in particular writing HTML/CSS/JavaScript code, using the WikiNEXT API for creating pages, talking to external or internal data sources was very simple. The major issue raised by users was in the writing and debugging of SPARQL queries (the syntax itself, and the need to escape some characters when used in JavaScript code was problematic to some testers). We rapidly wrote some tools to help for this particular task (a query composer with auto completion, a toolbox with ready to go requests, history of requests used by the user, etc.) These helper applications have all been written as WikiNEXT applications 35 . Larger-scale experiments in classrooms with master students from the University of Nice who follow the semantic web course, and with students who follow the HTML5 W3C online course will be conducted next year36 .

Discussion and Conclusions

WikiNEXT has been written from scratch using emerging technologies, as a reaction to the difficulties to learn and teach semantic web/linked data programming, also for writing documents that integrate data from the web of data. In particular it started as a front end for DBPedia.fr.

We developed WikiNEXT after writing two semantic wikis since 2005 (SweetWiki [START_REF] Buffa | SweetWiki: A semantic wiki[END_REF] and SweetDeki [START_REF] Buffa | ISICIL: Semantics and Social Networks for Business Intelligence[END_REF]); from these previous experiences we wanted to develop a semantic wiki that structures its content and maintains a global knowledge base from information added by different users and also keeps an upto-date value of changing data in a RDF store in a "more synchronous way". But we also designed WikiNEXT as a "programmable wiki", a wiki in which we can write documents with an application layer. The small applications written in this layer can be written using standard HTML/CSS/JavaScript, and can be shared, cloned, improved collaboratively. The applications can also be self-documented as they are parts of a wiki document. We also designed an API and protocols for keeping a tight, synchronous link between the data embedded in the pages and the databases on the server side. We showed how WikiNEXT can automatically create and/or update pages with metadata by using applications into the pages and templates, when other wikis such as Semantic Media Wiki uses extensions or plugins for this purpose. We have proposed also a new approach of combining web 2.0 technologies with semantic web technologies to support software development. The motivations of this effort, as well as related work and the research context were introduced, and the WikiNEXT API was described, in terms of provided functionalities and architecture. A case study was also presented. This paper introduces a real experience of the combination of collaborative technology (Wiki), knowledge technology (Semantic web) and web2.0 technologies to support software development. WikiNEXT will be used at the University of Nice and by the w3devcampus, the online platform that hosts the HTML5 W3C course.

Figure 1 :

 1 Figure 1: WikiNEXT page editors, the different tabs show the HTML editor, the JavaScript editor (current view) and the library management tab.

Figure 2 :

 2 Figure 2: WikiNEXT page model.

Figure 3 :

 3 Figure 3: WikiNEXT articles contain metadata that describe the articles themselves (title, date, author, etc.) and metadata that describes the articles' content.

Figure 4 :

 4 Figure 4: City application, here we see the form that will request DBPedia.org and display the results in the local page or in a new page.

Figure 5 :

 5 Figure 5: WikiNEXT Architecture.

  var pagedata = {}; pagedata.city = city; // dbpedia result article = wikinextHelper.city(pagedata);

Figure 6 :

 6 Figure 6: Results from DBPedia, default rendering without using a template. We call this rendering mode "sparqlOnFly".

Figure 7 :

 7 Figure 7: Results using a template.

Figure 8 :

 8 Figure 8: Map view of all cities in WikiNEXT

Figure 9 :

 9 Figure 9: Source code of SPARQL query to find all cities. We see the use of the WikiNEXT API line 36, and the use of templates for displaying the results (lines 42-55), see also Figure 10 for the HTML part of the template. The final rendered page is shown in Figure 7.

Figure 10 :

 10 Figure 10: Template used for generating the results shown in Figure 7.

Figure 11 :

 11 Figure 11: Search result example

http://twiki.org, http://foswiki.org, http://tiki.org, http://xwiki.org, https://atlassian.com/fr/software/confluence, http://mindtouch.com

See http://www.w3devcampus.com

https://moodle.org/

http://www.w3.org/TR/rdf-sparql-query/, the W3C standard for querying the web of data.

http://www.mindtouch.com/, uses DekisScript, a language inspired by JavaScript, for writing code within the pages.

http://velocity.apache.org/

http://groovy.codehaus.org/

http://tools.ietf.org/html/rfc6455

http://www.w3.org/TR/sparql11-query/

http://www.projecthalo.com/

http://www.inf.puc-rio.br/~psw12/, numerous papers details the

http://fr.wikipedia.org/wiki/Git

See for example http://designbeep.com/2011/06/28/18-useful-webbased-code-editors-for-developers/ for a list of such online services. More appear every month.

[START_REF] Meilendera | Semantic wiki engines: a state of the art[END_REF] https://scraperwiki.com/

http://d3js.org

https://github.com/pavel-arapov/wikinext

http://wikinext.gexsoft.com

http://schema.org/ and http://schema-rdfs.org have been created by the main search engine vendors and propose several RDF/S schemas for representing persons, articles, etc.

http://www.w3.org/TR/xhtml-rdfa-primer/ and http://www.w3.org/TR/rdfa-lite/

http://mongodb.org

http://nodejs.org

http://www.json.org, a text based notation for JavaScript objects.

http://mustache.github.io

http://wikinext.gexsoft.com/wiki/52304bb57a61be9c29000010

http://wimmics.inria.fr

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LGIT Clarisse Bordes

Laurence Jouniaux

Michel Dietrich

Jean-Pierre Pozzi

Stéphane Garambois

Laboratory measurements of seismo-magnetic conversions in fluid-filled Fontainebleau sand

Seismic wave propagation in fluid-filled porous materials induces electromagnetic effects due to small relative pore-fluid motions. In order to detect the seismo-magnetic couplings theoretically predicted by Pride (Phys. Rev. B, 50:15678-15695, 1994), we have designed a small-scale experiment in a low-noise underground laboratory which presents exceptional electromagnetic shielding conditions. Our experiment included accelerometers, electric dipoles and induction magnetometers to characterize the seismoelectromagnetic propagation phenomena. To assess the electrokinetic origin of the measured electric and magnetic fields, we compared records obtained in dry and fluid-filled sand. Extra care has been taken to ensure the mechanical decoupling between the sand column and the magnetometers to avoid spurious vibrations of the magnetometers and misinterpretations of the recorded signals. Our results show that seismo-electric and seismomagnetic signals are associated with different wave propagation modes, thus emphasizing the electrokinetic origin of these effects.

1

Introduction

Observations of transient electromagnetic phenomena accompanying the seismic wave propagation in fluid filled porous media date back at least to the work of [START_REF] Ivanov | Seismic-electric effect of second kind[END_REF]. [START_REF] Frenkel | On the theory of seismic and electroseismic phenomena in a moist soil[END_REF] gave the first quantitative explanations of these phenomena in term of electrokinetic effects at the pore scale until [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF] developed a complete theory which prompted further studies.

Early and pionneering work by [START_REF] Martner | The electroseismic effect[END_REF] and [START_REF] Thompson | Geophysical applications of electrokinetic conversion[END_REF] and more recent studies by [START_REF] Takeuchi | Observations of earthquake waves by the vertical earth potential difference method[END_REF], [START_REF] Mikhailov | Using borehole electroseismic measurements to detect and characterize fractured (permeable) zones[END_REF], and [START_REF] Garambois | Seismoelectric wave conversions in porous media: Field measurements and transfer function analysis[END_REF] have concentrated on field measurements. Laboratory measurements were notably performed by [START_REF] Zhu | Experimental studies of seismoelectric conversions in fluid-saturated porous media[END_REF], and [START_REF] Zhu | Seismoelectric and seismomagnetic measurements in fractured borehole models[END_REF] whereas numerical simulations were performed by [START_REF] Haartsen | Electroseismic waves from point sources in layered media[END_REF], [START_REF] Garambois | Full waveform numerical simulations of seismoelectromagnetic wave conversions in fluid-saturated stratified porous media[END_REF] and [START_REF] White | Asymptotic theory of electroseismic prospecting[END_REF].

Two kinds of seismo-electromagnetic effects are to be distinguished. The dominant contribution we are adressing in this paper corresponds to electrical and magnetic coseismic fields accompanying the body and surface waves. The second kind is generated at physico-chemical properties contrasts and consists of independantly propagating electromagnetic waves.

Seismo-electromagnetic studies have generally concentrated on the measurements of electrical fields as they require only a simple instrumentation. The investigation of seismo-magnetic fields has received much less attention mainly because of the high level of electromagnetic noise affecting the magnetic measurements. In order to minimize these disturbances, we have designed a laboratory experiment within the ultra-shielded chamber of the LSBB low-noise laboratory located in Rustrel, southern France. This paper describes the experimental apparatus as well as the first results (seismic, electric and magnetic responses) measured in homogeneous Fontainebleau sand. We show that seismo-magnetic conversions are weak but nevertheless measurable. Moreover, the different apparent velocities characterizing the seismo-electric and seismo-magnetic events emphasize that they are associated to different propagation modes.

Experimental apparatus

The LSBB facilities were originally an underground launching center for the ground-based component of the French strategic nuclear defense. A characterization of the electromagnetic shielding, performed by [START_REF] Gaffet | Simultaneous seismic and magnetic measurements in the low-noise underground laboratory (lsbb) of rustrel, france, during the 2001 january 26 indian earthquake[END_REF], using a SQUID magnetometer showed that the noise level is below 2 fT/ √ Hz above 10 Hz.

Our experimental apparatus was located within the shielded chamber and consisted of a porous sample, a seismic source and sensors. The whole experiment including the triggering of the mechanical source and the data acquisition was remotely controlled from outside the chamber to suppress electromagnetic perturbations from the instruments. All measurements were performed with a 24 bit seismic recorder (Geometrics StrataVisor NZ) using a 21 µs time sampling rate.

2.1 Seismo-electric and seismo-magnetic measurements.

Our experiments were performed with two 1 m high and 8 cm diameter vertical Plexiglas columns filled with Fontainebleau sand (figure 1). This sand contains 99 % of silica with grain size smaller than 300 µm. The measured permeability of the sand is 5.8 10 -12 m 2 and its bulk density 1.7727 10 3 kg/m 3 , its electrical resistivity is 22 kΩ.m, and the water conductivity is 3.1 mS/m with a pH of 6.55 at 20.5 • C. The sand was compacted by vibrating the column during filling in order to minimize the pore space and high frequency seismic wave attenuation. The seismic velocities in the partially saturated sand can be estimated by considering reasonable values of the bulk frame modulus (1.3 10 9 Pa) and shear frame modulus (1.4 10 9 Pa). By using these values in relations given by [START_REF] Pride | Electroseismic wave properties[END_REF] the computed P and S wave velocities are respectively equal to 1300 and 870 m/s.

The first column was equipped with ten unpolarizable electrodes (silver rods and porous ceramics) spaced 10 cm apart along the column generatrix.

These electrodes were previously used for streaming potential measurements and are described in [START_REF] Guichet | Streaming potential of a sand column in partial saturation conditions[END_REF]. We chose to use a common reference electrode at the bottom end of the column. Electrical measurements (mV) are normalized by the dipole spacing to provide equivalent electrical field (mV/m).

This column was also equipped with four accelerometers fixed on the outer Plexyglas surface.

The second column was especially used for the seismo-magnetic measurements which are much more sensitive to external perturbations. Even if it is reduced within the capsule, we have to take into account the constant magnetic field. Indeed, the motions of the magnetometers may generate induction effects resembling coseismic effects and masking the seismo-magnetic signals of interest. We designed a suspension system for the seismic source on the capsule ceiling to avoid the transmission of mechanical vibrations to the magnetometers. Moreover, the magnetometers were fixed on a separate stand, and were isolated from ground vibrations by a soundproofing material. However, test accelerometers fixed on the magnetometers showed that residual vibrations arrive 5 ms after the seismic impacts. These disturbances appear later than the seismic and seismo-electromagnetic signals but are nevertheless in the time window considered (-1 ms to 10 ms).

The upper magnetometer can be moved into seven locations which exactly correspond to the electrode positions in term of source-receiver spacing (figure 1). The two missing locations correspond to the column fastening. The lower magnetometer is used as a common reference similarly to the electrical measurements, in order to cancel synchronous disturbances such as accelerometer radiation.

The theory of [START_REF] Pride | Governing equations for the coupled electromagnetics and acoustics of porous media[END_REF] stipulates that in a homogeneous infinite fluidsaturated porous medium, the coseismic seismo-electric fields are traveling with longitudinal P -waves whereas seismo-magnetic signals are associated with transverse S -waves. For porous cylinders, we have to consider global and local deformations and their propagation in the structure. The maximum displacements due to a vertical seismic excitation are associated with extensional modes. In the elastic case, the extensional wave velocity is derived from Young's modulus E and total density ρ (V ext = E/ρ=1300 m/s) and is close to the P -wave velocity. Thus, waves propagating at a velocity of 1300 m/s will be associ-ated with P and extensional modes whereas waves traveling at approximately 870 m/s will be associated with S waves. We used an induction magnetometer whose sensitivity on the radial and tangent components is quite constant in the 100 Hz to 2 kHz range.

EM noise free pneumatic seismic source

Piezoelectric seismic sources are often used for seismo-electric laboratory measurements [START_REF] Ageeva | E-effect in rocks[END_REF][START_REF] Zhu | Experimental studies of seismoelectric conversions in fluid-saturated porous media[END_REF]. These sources, however, are not well suited for seismo-magnetic measurements because of their intrinsic electromagnetic radiation. In order to minimize the magnetic perturbations generated by the seismic excitation, we built a pneumatic system (figure 2) capable of generating a large number of impacts within a few minutes. This feature allows us to improve our data by stacking successive recordings.

An accelerometer placed on the granite cylinder records the source time function. The latter is rather impulsive and has a broad-band spectrum ranging from 100 Hz to 10 kHz. To compare the records obtained from different shots, the signals are all normalized with respect to a reference excitation of 1000 m/s 2 ( 100 g).

Results

Since

Homogeneous dry sand

The first arrivals of the accelerometric signals recorded in dry sand show an apparent velocity of 1201 ± 85 m/s in the 700 Hz to 3 kHz frequency range (figure 3a). However, panels 3b and 3c show that the seismic wave propagation in homogeneous dry sand does not produce any coherent electromagnetic fields. The electrodes only pick up instrumental noise with amplitudes lower than 0.4 mV/m for an impact of 1000 m/s 2 (figure 3b). Nevertheless, the trace located nearest to the source shows a weak signal (1.4 mV/m/1000 m/s 2 ) probably associated with a piezoelectric effect due to the excitation of quartz grains in perfectly resistive conditions. Similarly, the magnetic signals (figure 3c) display low frequency noise caused by internal electronic disturbances of the magnetometers.

Fluid-filled homogeneous sand

The seismic records presented in figure 3d are slightly modified by the presence of water. The lower frequency content of the signals seen in figure 3d, as compared to figure 3a, can be explained by the fact that in [START_REF] Biot | Theory of propagation of elastic waves in a fluid-saturated porous solid: I. low frequency range[END_REF] theory, high frequencies are strongly attenuated due to fluid flow at the pore scale.

The similarity of the apparent velocities of the first seismic arrivals in dry and saturated conditions is consistent with a fluid saturation lower than 80 % according to the [START_REF] Knight | A laboratory study of the dependence of elastic wave velocities on pore scale fluid distribution[END_REF] measurements in rocks.

When the sand is filled with water, electric and magnetic fields records are strongly modified. In particular, we observe that the amplitudes of the Although rather weak, seismo-magnetic signals can be detected with sensitive induction magnetometers but would be difficult to measure outside of the ultra shielded chamber. Nevertheless, transverse modes could be enhanced by considering a horizontal seismic excitation that would generate stronger seismo-magnetic amplitudes. Our results emphasize the complementary nature of seismo-electric and seismo-magnetic measurements to estimate the properties of porous media especially in boreholes.

  our objective is to demonstrate the existence of seismo-electromagnetic fields due to fluid-grain interactions, we compared the electric, magnetic and seismic responses measured in dry and fluid-filled homogeneous sand. The signals presented in figure 3 were obtained by stacking 10 accelerometric, 10 seismo-electric, and 100 seismo-magnetic records. The panels displayed in figure 3 respectively present the seismic (top), seismo-electric (middle) and seismomagnetic (bottom) responses obtained in dry sand (left) and fluid-filled sand (right). The spectra of seismic and electric signals were computed from the complete time sequences whereas the spectra of the magnetic signals were limited to the (-1 ms to 4 ms) time window.

  figure 3e) are 30 times larger than the electrical noise level in dry sand. The contributions of seismo-electric conversions are visible in both the time section and corresponding amplitude spectrum in the 200 Hz to 2 kHz range.By contrast, the maximum amplitudes of the seismo-magnetic signals are only 3 times as high as in dry sand. Consequently, seismo-magnetic signals are barely visible in the associated amplitude spectrum (figure3f). However, there is clear evidence of coherent arrivals in the time section. The presence of these events indicates some weak but non-zero seismo-magnetic coupling.

Figure 1 :Figure 2 :

 12 Figure1: Experimental apparatus for seismo-electric and seismo-magnetic meathe column on the left is equipped with 10 unpolarizable electrodes and 4 accelerometers; the column on the right is used for seismo-magnetic measurements only. Magnetometers are fixed on an independent stand to avoid the transmission of disturbing vibrations.

Figure 3 :

 3 Figure 3: Seismic, seismo-electric and seismo-magnetic signals measured in homogeneous Fontainebleau sand normalized with respect to a 1000 m.s -2 vertical excitation. Amplitude spectra corresponding to dry conditions are shown in grey in the right-hand side panels for comparison. Apparent velocities have been estimated from the linear regression of first arrivals time.

Acknowledgements

This research was supported by "ACI Eau et Environnement" of the French Ministry of Research. We are grateful to G. Waysand and S. Gaffet for helpful discussions and autorization to perform this experiment in the LSBB laboratory.

We thank M. Auguste, D. Boyer, A. Cavaillou, G. Clerc, R. Guiguet and Y.

Orengo for their technical help.


C Etchegaray
email: christele.etchegaray@math.u-psud.fr

B Grec

B Maury

N Meunier

L Navoret

An integro-differential equation for 1D cell migration

Introduction

Cell migration is a fundamental biological phenomenon involved for example in development, wound healing, cancer and immune response. Understanding its key features is therefore a burning issue. Some cells can move on an adherent substrate by a crawling process, where motion comes from the formation of finger-like extensions named filopodia that adhere to the substrate for some time. When the cell contracts, non-adherent filopodia retract, whereas adherent ones exert forces that induce motion. We refer to [START_REF] Ananthakrishnan | The Forces Behind Cell Movement[END_REF] for a complete description of cell crawling.

When a cell is set on a flat homogenous substrate, it performs a randomlike motion. However, it can also become polarized and move in a preferential direction. How this direction is chosen is a question that is still driving many experimental and modeling efforts. In [START_REF] Caballero | Protrusion Fluctuations Direct Cell Motion[END_REF], a non-homogenous substrate impose geometrical constraints that are sufficient to direct 1D cell motion. This paper focuses on the 1D motion brought forth by the filopodial activity.

We introduce a simplified model of 1D cell migration relying on the filopodial activity. In what follows, the substrate is supposed to be flat and homogenous, but more complex settings could also be described. Let us consider the center of mass of the cell, whose position at time t is denoted x(t) ∈ R. Force equilibrium leads to

C dx dt (t) = -F ℓ (t) + F r (t), (1.1)
where F r ≥ 0 (resp. F ℓ ≥ 0) is the force exerted by filopodia located on the right (resp. on the left) of the cell and C is the friction parameter, that can be set equal to 1nN.h.µm -1 [START_REF] Wong | Finite element analysis of the effects of focal adhesion mechanical properties and substrate stiffness on cell migration[END_REF]. We can now focus on the forces F r,ℓ . Following biological knowledge, we assume that the forces exerted by filopodia on the cell at time t depend on:

• densities of filopodia sent to the right and left, denoted by ψ r,ℓ > 0,

• their existence time, fixed by the lifetime function P : R + → R + , • the force f r,ℓ exerted by one filopodium on the cell, related to its orientation. Moreover, we assume that f r,ℓ = f r,ℓ (x(t ′ ), x(t)) depends on the positions of both the tip of the filopodium, related to the cell position at creation time t ′ , and the actual cell position.

Consequently, equation (1.1) rewrites as an integro-differential equation

dx dt (t) = t 0 P(a) ψ r f r (x(t -a), x(t)) -ψ ℓ f ℓ (x(t -a), x(t)) da, (1.2)
x(0) = x 0 , where ψ r,ℓ are positive constants, x : R + → R, and f r,ℓ : R 2 → R. Let us assume for simplicity that x 0 = 0. Problem (1.2) can be treated more or less easily depending on the force functions f r,ℓ . In this work, we shall first investigate one case of non-linear elastic force, where only existence and uniqueness of a solution can be proved. Then, we shall consider a simplified case of linear force functions, where a linear Volterra equation can be obtained. We shall see how this formalism allows us to get more information on the sign, boundedness and asymptotic behaviour of the solution in general, as well as explicit solutions for some special cases.

Non-linear force functions

Let us start with the force functions

f r (y, x) = k [ℓ -(x -y)] + , and f ℓ (y, x) = k [ℓ -(y -x)] + ,
where [.] + denotes the positive part function and k, ℓ ∈ R + are two constants. Taking f r (x(t-a), x(t)) and f ℓ (x(t-a), x(t)), it corresponds to the hypothesis of filopodia having a constant size ℓ, and exerting elastic forces as long as the cell at position x(t) has not reached their tips x(t -a) ± ℓ. Equation (1.2) now writes dx dt

(t) = k t 0 P(a) ψ r [ℓ + x(t -a) -x(t)] + -ψ ℓ [ℓ + x(t) -x(t -a)] + da.
(1.3)

Existence and uniqueness

We prove the following result :

Theorem 1. For P ∈ L 1 (R + ), there exists a unique solution x ∈ C 1 (R + , R) of (1.3).

Proof. After integration, equation (1.3) writes

x(t) = k t 0 s 0 P(a) ψ r x(s -a) + ℓ -x(s) + -ψ ℓ x(s) -x(s -a) + ℓ + dads =: Φ(x)(t)
with

Φ : C([0, T ], R), ∞ -→ C([0, T ], R), ∞ x -→ Φ(x) = (t → Φ(x)(t)),
for some T ≥ 0. We are looking for existence and uniqueness of a fixed point for Φ. Let us construct a sequence

(x n ) n≥0 in C([0, T ], R) such that x 0 ≡ x 0 , x n+1 = Φ(x n ) ∀n ≥ 0. As [0, T ] is compact, C([0, T ], R),
∞ is a Banach space and we can use the Banach fixed-point theorem. All we need to show now is that Φ is a contraction mapping. Considering (y, z) ∈ C([0, T ], R), ∞ 2 and denoting g s,a (y) = y(s -a) + ℓ -y(s), and h s,a (y) = y(s) -y(s -a) + ℓ, we have

Φ(y) -Φ(z) ∞ = sup t∈[0,T ] k t 0 s 0 P(a) ψ r [g s,a (y)] + -[g s,a (z)] + -ψ ℓ [h s,a (y)] + -[h s,a (z)] + dads ≤ kT sup s∈[0,T ] s 0 |P(a)| × ψ r [g s,a (y)] + -[g s,a (z)] + + ψ ℓ [h s,a (y)] + -[h s,a (z)] + da, since ψ r,ℓ ≥ 0. Denote ψ := ψ r + ψ ℓ . Now, for (A, B) ∈ R 2 , the inequality [A] + -[B] + ≤ A -B holds, leading to Φ(y) -Φ(z) ∞ ≤ kT sup s∈[0,T ] s 0 ψ |P(a)| (y -z)(s -a) -(y -z)(s) da, ≤ 2kT ψ P L 1 (R+) y -z ∞ .
For T small enough such that 2kψ P L 1 (R+) T < 1, we deduce that Φ is a contraction mapping. As a consequence of the Banach fixed-point theorem, there exists a unique x ∈ C([0, T ], R) which is solution of (1.3). Iterating the same reasoning on time intervals of size T , one can extend this result to prove that (1.3) admits a unique solution x ∈ C(R + , R). Finally, using (1.3), it is clear that x ∈ C 1 (R + , R), and this concludes the proof.

Numerical simulations

We consider the lifetime function P : a → e -a . This means that a density of filopodia will exponentially decrease with time, as more and more filopodia will have disappeared. For the filopodia's length, we use an experimental value from [START_REF] Caballero | Protrusion Fluctuations Direct Cell Motion[END_REF]. Moreover, we impose a bias on the densities of filopodia (ψ r > ψ ℓ ).

Figure 1.2 represents the cell position and velocity over time computed with an explicit Euler time discretization and a rectangle integration method of equation (1.3). What can be observed is that the bias in the produced forces seems to lead to a non-zero asymptotic velocity. Further simulations with different parameter values and/or lifetime functions confirm this tendency.

Linear forces

The presence of the positive part function in the previous case prevents getting analytical properties of the solution. In this section, we will take the following linearized forces functions:

f r (x(t -a), x(t)) = k (x(t -a) + ℓ -x(t)) , f ℓ (x(t -a), x(t)) = k (x(t) -x(t -a) + ℓ) ,
with k, ℓ ∈ R + . This assumption is less relevant from the modelling point of vue, since if the cell overtakes the tip of a filopodium, then it will experience a force in the opposite direction. However, for k small enough, and an appropriate lifetime function, we can assume that the cell is slow enough so that it does not reach any existing filopodium tip.

Equation (1.2) can be written as a linear Volterra equation, which will lead to more analytical results.

Linear Volterra equation formalism

Let us rewrite equation (1.2):

v(t) = k t 0 P(a) ψ r (x(t -a) + l -x(t)) -ψ ℓ (x(t) -x(t -a) + ℓ) da = kl(ψ r -ψ ℓ ) t 0 P(a)da + kψ t 0 P(a) (x(t -a) -x(t)) da.
(1.4)

Denoting Q : t → t 0 P(a)da and integrating by parts, we obtain

v(t) = kℓ(ψ r -ψ ℓ )Q(t) + kψ t 0 Q(a)v(t -a)da -Q(t)x(t) ,
since Q(0) = 0 and x(0) = 0. After the change of variable s = t -a, we get

v(t) = f (t) -kψ t 0 (Q(t) -Q(t -s))v(s)ds, (1.5) with f (t) = kℓ(ψ r -ψ ℓ )Q(t). (1.6)
which is a linear Volterra integro-differential equation on v.

Existence and uniqueness of a solution

With similar arguments to Theorem 1, we can prove the following property:

Theorem 2. For P ∈ L 1 (R + ), equation (1.5) admits a unique solution v ∈ C(R + , R).
Remark 3 (The resolvent formalism) Let us define the operator

h ⋆ v : t → +∞ 0 h(t, s)v(s)ds.
Equation (1.5) can then be written as a convolution-like equation:

v(t) + (h ⋆ v)(t) = f (t), with h(t, s) = kψ(Q(t) -Q(t -s))1 [0,t] (s).
Existence and uniqueness of a solution can be proved by showing that h is a Volterra kernel of L ∞ type. For more details, we refer to [START_REF] Gripenberg | Volterra Integral and Functional Equations[END_REF].

Sign and boundedness property

We now prove a result showing how important the function f is in controlling the migration. Indeed, it captures no less than the range of forces exerted with k, aging, and the potential asymmetry ψ r -ψ ℓ in the formation of filopodia.

Theorem 4. If P is positive and decreasing, then the solution to equation

(1.5) satisfies ψ r ≥ ψ ℓ ⇒ ∀t ≥ 0, v(t) ∈ [0, f (t)], ψ r ≤ ψ ℓ ⇒ ∀t ≥ 0, v(t) ∈ [f (t), 0].
Proof. First, let us assume that ψ r ≥ ψ ℓ . As a consequence, ∀t ≥ 0, f (t) ≥ 0 and f ′ (t) ≥ 0. By derivation of (1.5), we obtain

v ′ (t) = f ′ (t) -kψ(Q(t) -Q(0))v(t) -kψ t 0 (P(t) -P(t -s))v(s)ds,
with f ′ (t) = kℓ(ψ r -ψ ℓ )P(t). Suppose there exists t * such that ∀t < t * , v(t) > 0 and v(t * ) = 0, then

v ′ (t * ) = f ′ (t * ) -kψQ(t * )v(t * ) -kψ t * 0 (P(t * ) -P(t * -s))v(s)ds
is positive (since all the terms are positive). Consequently, ∀t ≥ 0, v(t) ≥ 0. This implies that x(t -a) -x(t) ≤ 0, ∀t ≥ a ≥ 0. Going back to the equivalent equation (1.4), this shows that ∀t ≥ 0, v(t) ≤ f (t). Now, let us assume that ψ ℓ ≥ ψ r , which means that ∀t ≥ 0, f (t) ≤ 0 and f ′ (t) ≤ 0. In a similar way, if there exists t * such that ∀t < t * , v(t) < 0 and v(t * ) = 0, then v ′ (t * ) is negative. And considering again (1.4), we easily show that ∀t ≥ 0, v(t) ≥ f (t), which concludes the proof.

Asymptotic velocity

We now give an expression for the asymptotic velocity of the cell. Here again, f has a crucial importance. The proof of the following result is similar to the one done in a forthcoming work [GrMaMeNa], and we do not repeat it here.

Theorem 5. Let v be the solution of (1.5), and denote γ = lim t→+∞ f (t). Assume that v is uniformly continuous on R + . Then,

v(t) -→ t→+∞ v ∞ =      γ 1+kψ +∞ 0 aP(a)da if a → aP(a) ∈ L 1 (R + ), 0 if a → aP(a) / ∈ L 1 (R+).
Having two different cases can be interpreted as follows: if the mean lifetime of filopodia is finite, then the cell is permanently escaping from the action of older forces. As a consequence, if ψ r -ψ ℓ = 0, it can get off its position all the time. However, if the mean lifetime of filopodia is infinite, all of them exert elastic forces on the cell, which will be stabilized in finite time.

Particular cases

Some choices of function P can give more explicit information on the solution.

Infinite existence time of forces (P ≡ 1).

Taking P ≡ 1, we are considering elastic forces that never disappear. Here, P does not fulfill the hypothesis of Theorem 2, but Theorem 5 applies. Equation (1.5) writes

v(t) = kℓ(ψ r -ψ ℓ )t -kψ t 0 sv(s)ds,
and can be solved after derivation with the variation of constants method, to give

v(t) = v(0)e -kψt 2 /2 + ℓ kπ 2 (ψ r -ψ ℓ ) e -kψt 2 2 1 -e -kψt 2 2 .
(1.7)

Figure 1.3 represents the solution v as well as the corresponding forcing function f . As expected, the cell is stabilized in finite time. Moreover, we can observe numerically the sign and boundedness property (given in Theorem 4), where in this case the f (t) bound is optimal.

It is easy to check analytically the convergence of v to v ∞ = 0 since we have the following equivalence: We assume now that P(a) = e -a , which was the function chosen in Section 1.2.2. All the results demonstrated before apply. However, we can actually directly solve the equation. Noting that Q(t) = 1-e -t , equation (1.5) becomes

v(t) ∼ t→+∞ ℓ kπ 2 (ψ r -ψ ℓ )e -kψt 2 /4 .
v(t) = kℓ(ψ r -ψ ℓ )(1 -e -t ) -k(ψ r + ψ ℓ )e -t A(t), (1.8) with A(t) = t 0 (e s -1)v(s)ds.
Proposition 1. The solution to (1.8) is given by: v(t) = kℓ(ψ r -ψ ℓ ) 1 -e -t -k 2 l(ψ r -ψ ℓ )ψe -(kψ+1)t+kψ-kψe -t J(t), (1.9)

with J(t) = t 0 (e s + e -s -2)e kψ(s-1+e -s ) ds.

Proof. Deriving A with respect to time leads to:

A ′ (t) = kℓ(ψ r -ψ ℓ )(e t + e -t -2) -kψ(1 -e -t )A(t).
Now, by the variation of constant method, we find

A(t) = kℓ(ψ r -ψ ℓ )e -kψ(t-1+e -t )
t 0 (e s + e -s -2)e kψ(s-1+e -s ) ds, leading to expression (1.9).

As a consequence, an asymptotic equivalent of the solution can be given.

Theorem 6. The following equivalence holds:

v(t) ∼ t→+∞ kℓ(ψ r -ψ ℓ ) 1 - kψ kψ + 1 - kψ kψ -1 e -2t + 2e -t , (1.10)
and v converges to the asymptotic velocity

v ∞ := kℓ(ψ r -ψ ℓ ) 1 - kψ kψ + 1 .
Proof. Using the following equivalence

t 0 e αs ds ∼ t→+∞ e αt α ,
we easily obtain

J(t) ∼ t→+∞ e -kψ e (kψ+1)t kψ + 1 + e (kψ-1)t kψ -1 -2 e kψt kψ .
Considering expression (1.9), we have :

v(t) ∼ t→+∞ kl(ψ r -ψ ℓ ) 1 -kψe -(kψ+1)t e (kψ+1)t kψ + 1 + e (kψ-1)t kψ -1 -2 e kψt kψ ,
which leads to the result.

We can then deduce that the asymptotic behaviour of the cell depends on the range of filopodial forces k, on the global filopodial activity ψ, and on the asymmetry ψ r -ψ ℓ . Moreover, the bigger k and ψ are, the faster the cell velocity reaches equilibrium. The non-trivial equilibrium is a consequence of the lifetime function that lets newer forces lead motion, whereas the older ones are "silenced". The initial asymmetry is then maintained over time. On Figure 1.4, a numerical simulation illustrates this behaviour.

Constant existence time (P(a) = 1 [0,τ ] (a))

Now, let us look at P(a) = 1 [0,τ ] (a) with τ > 0, meaning that all filopodia exert forces during the same finite amount of time. For t ≥ 0, we have

Q(t) = t if t < τ, τ if t ≥ τ.
Existence and uniqueness of a continuous solution to (1.5) comes from Theorem 2. Moreover, we can find an explicit solution for t ≤ τ, and bounds for the solution for t ≥ τ . Equation (1.5) then writes

v(t) = kℓ(ψ r -ψ ℓ )t -kψ t 0 sv(s)ds, for t ≤ τ, (1.11) v(t) = kℓτ (ψ r -ψ ℓ ) -kψ t t-τ v(s)(τ -(t -s))ds, for t ≥ τ.
(1.12) Theorem 7. The unique solution to equations (1.11)-(1.12) satisfies

v(t) = ℓ(ψ r -ψ ℓ ) kπ 2ψ (1 -e -kψ 2 t 2 ) for t ≤ τ, (1.13) v(τ ) exp -kψ (t 2 -τ 2 ) 2 ≤ v(t) ≤ kℓ(ψ r -ψ ℓ ) for t ≥ τ. (1.14)
Proof. Let us first study the case where t ≤ τ . By derivation of (1.11), we have:

v ′ (t) = kℓ(ψ r -ψ ℓ ) -kψtv(t),
and the variation of constant method leads to expression (1.11).

Let us now consider the case where t ≥ τ . After a change of variable, equation (1.12) becomes

v(t) = kℓτ (ψ r -ψ ℓ ) -kψ τ 0 (τ -s)v(t -s)ds, (1.15) = kℓτ (ψ r -ψ ℓ ) -kψ t 0 (t -s)v(t -s)ds + h(t), (1.16)
with

h(t) = kψ t τ (t -s)v(t -s)ds + kψ τ 0 (t -τ )v(t -s)ds.
Since P is positive and decreasing, we deduce from Theorem 4 that ∀t ≥ 0, v(t) ≥ 0. Hence, we know that h ≥ 0 on [τ, +∞[. Moreover, derivating h in t, we obtain

h ′ (t) = kψ t τ d dt ((t -s)v(t -s))ds + τ 0 d dt ((t -τ )v(t -s))ds , = kψ ((t -τ )v(t) + (x(t) -x(t -τ ))) ≥ 0, as v ≥ 0. Then, derivating equation (1.16) leads to v ′ (t) ≥ -kψtv(t), from which we deduce v(t) ≥ v(τ ) exp -kψ (t 2 -τ 2 ) 2 ,
leading to the first inequality. Moreover, as v ≥ 0, the second one is obtained from equation (1.15), and this concludes the proof.

Conclusions and perspectives

In this paper, we have introduced a simple deterministic model of 1D cell migration, based on the filopodial activity of the cell. It describes the formation of antagonist elastic forces by filopodia on each side of the cell. This model is not able to describe realistic trajectories, as the filopodial activity is taken constant, but it relates explicitly filopodial statistics to the cell velocity and asymptotic behavior, and hence represents a first step in the global description of cell trajectories from filopodial activity.

In this work, we have studied a realistic case where filopodia stop exerting a force as soon as the cell has overtook their tips. In this case, the highly nonlinear force prevent us from getting more than an existence and uniqueness result.

The case of linear elastic forces is richer, as it gave more information about the sign, boundedness and asymptotic behaviour of the solution. It is important to keep in mind that the linear model is realistic only in a particular setting: if the cell is slow enough, and filopodia's lifetime short enough, then they won't be reached by the cell. Typical velocity and filopodium lifetime are closely related to the cell type and experimental setting. Indeed, considering the force kℓ exerted by a filopodium of length ℓ on the substrate, it is known that ℓ is variable among cell types. Moreover, k highly depends on the rigidity of the substrate: the more it is rigid, the larger the forces are [START_REF] Lo | Cell Movement Is Guided by the Rigidity of the Substrate[END_REF].

Another key-player in the filopodial forces is the adhesiveness of the substrate, that scales how strong it is coupled to filopodia, hence how large forces will be. However, a very adherent substrate is also less likely to let go of filopodia during the contraction of the cell, leading to a longer lifetime for them. This results in a bell-shaped curve relating velocity and adhesiveness, as described in [START_REF] Palecek | Integrin-ligand binding properties govern cell migration speed through cell-substratum adhesiveness[END_REF]. As a consequence, it is likely that for a substrate of low (or very large) adhesiveness and low rigidity, cells velocity would be low enough so that the linear model fits with experimental conditions. This firststep model describing filopodial activity and trajectories is simple enough to give analytical information about the cell velocity, but still rich enough to be compared to different kinds of experimental 1D migration assays. In future works, it will be crucial to consider non constant densities of filopodia, to take into account the effect of motion itself on the filopodial activity. This would probably lead to much more realistic trajectories, where changes of direction would be described.

Fig. 1. 1 .

 1 Fig. 1.1. Illustration of a moving cell, and picture of a fibroblast [MaSiBrYeMe08].

Fig. 1

 1 Fig. 1.2. Numerical simulation of a particle speed and trajectory during T = 10h, for dt = 10 -2 h, C = 1nN.h.µm -1 , k = 1nN.µm -1 , ℓ = 20.5µm, and (ψr, ψ ℓ ) = (1.5, 1).

Fig. 1 . 3 .

 13 Fig.1.3. Graph of t → v(t) and t → f (t) for P ≡ 1, with C = 1nN.h.µm -1 , ℓ = 20.5µm, k = 5nN.µm -1 , (ψr, ψ ℓ ) = (6, 4) and v(0) = 0.

Fig. 1. 4 .

 4 Fig. 1.4. Numerical simulation of the exact solution, its equivalent at infinity and the asymptotic velocity for C = 1nN.h.µm -1 , k = 1nN.µm -1 , ℓ = 20.5µm and (ψr, ψ ℓ ) = (1.5, 1).