The University Library and the Humanities: Historical and Contemporary Perspectives Anne Baillot Berlin, 15. 09. 2011 In this paper, I would like to address the central questions of the conference by presenting the specific case of University Libraries. University Libraries play an important role in the constitution and institutionalized transmission of knowledge. They display interesting phenomena related to the process of institutionalization of the Humanities and connect to the question of their extrainstitutional development. In their founding phase, they also relate strongly to the national culture of the place where they are founded. I will begin this paper with a reconstruction and an analysis of the early history of the University libraries of Berlin and of Chicago. I will then sketch some of today's orientations of both libraries. I hope to be able to show then that reconstructing the origins of these institutions can open perspectives for Humanities today and tomorrow.

Hitzig's Reading Room

When the Berlin University was founded, in 1810, Prussia was in a situation of great political distress due to the concessions it had to make to Napoleon. The decision to found a University in the capital city was neither easy nor natural. But the fact that the city had a series of institutes and collections, a solid hospital, two academies and consequently a pool of scholars already active pleaded in favor of Berlin. In times of post-war budget-cuts, mutualizing infrastructure was the core of the whole university project. 1 What was created was not a university as we understand it nowadays with all of its administration, rules and substructures. It lacked, among a lot of other things, a University Library attached to it. Wilhelm von Humboldt considered that the access granted to the Royal Library and its books stock was sufficient to the needs of students as well as of professors. 2 The Royal Library was within the hands of the Academy of Sciences located opposite to the University itself, and several members of the University staff were also Academy Members. But despite the geographical and scientific closeness of interests, the collaboration was not as successful as one could have expected.

There were in fact several problems connected to the absence of a proper University Library. First, the stock of the Royal Library was not intended for University use only: Academics would miss books borrowed by other users and the other users would miss the books used by Academics. Second, the Royal Library offered a book stock, but the reading room had ridiculously reduced opening hours (only a couple of hours within the week) 3 . And third, its stock didn't contain the books and reviews that were necessary to research and teaching at a top level. Although all Prussian publishers were in theory compelled to send in a certain amount of their new publications to the Royal Library, its stock was in fact not representative of what was at the forefront of science. Many of the publishers did actually not send the due books, and, most importantly, the scientific reviews were not part of the deal.

This was, as recent works of Anna Busch have shown,4 the main problem. Scientific reviews were the best and practically only way to connect to what was on the forefront of scholarship. Having access to them was necessary to anyone who wanted to keep updated concerning evolutions and debates, experiments and discoveries, editions and manuscripts newly found. The problem for students, and for professors too, was that they could not financially afford as many review subscriptions as their studies would require. Some reviews could be read in cafés, but not all and most of the time, these were more political or literary than scientific reviews.

The publisher and businessman Julius Eduard Hitzig soon realized that he could develop a fairly profitable business model that would fill in the gap. He created a reading room which professors and students could access for a decent subscription fee. They could find there the scientific reviews and the textbooks on the topics lectured at the University. Writing material was also at their disposal, so that they might be able to take notes they could take back home. The reading room was opened every day until dusk. Hitzig, who knew the professors, asked them well ahead of time what books would be of use for their next class, so that he had ordered and received them when the new semester started. It was also possible to issue queries for a specific scientific review or book. The subscriptions ended up being largely sufficient to provide all the necessary material and generated a substantial profit for Hitzig. In the first years, a vast majority of students and professors had subscribed.

Considering this success -not the least of it being the economic success it represented -it would have only be natural for the University to take over such a model. But it took the University another ten years to realize its own reading room -and what is more, one which students were not granted access to. 5 And it took yet another ten years for the University to get, 1831, a Library of its own. Before I come to the founding of the University Library itself, let me draw a couple of preliminary conclusions:

 First, we have to do with a situation in which a private person successfully takes over the role of an institution. This has to do with the fact that by 1810 intellectual networks had a long history of being structured by the Bildungsbürgertum, contrarily to the newly founded University, which lacked many of its institutional fundaments for several years. In my view, Humboldt did not as much found the Berlin University as hand over a couple of bricks. It is in fact the people who worked at the University and at the Ministry of Cult between 1810 and 1820/30 that really founded the Berlin University. This process, as we can see with the example of the Hitzig reading room, was supported by civil society.

 Second -and this has to do with Berlin intellectual networks as well -Hitzig's reading room was based on a constant exchange of information. Of course, it could only work because it developed on a small scale -pretty much everybody knew everybody in intellectual Berlin around 1810. But it is still remarkable that it was the collective input that allowed to optimize the exchange of information. In that sense, it is a very modern system.

Foundation of the University Library

Let me come now to the founding of the University Library itself. Because of its institutionalized dimension, it took a much slower and more complicated start than Hitzig's Reading Room. It had to be an institution that should last as long as the University itself and in that sense, it had to establish its legitimacy.

In 1817, Friedrich Wilken was appointed to the University of Berlin. He had served at the University of Heidelberg as a professor of oriental history and chief librarian of the University Library. It was him who founded and directed the University Library in Berlin too. Wilken urged Minister and King to create a specific University Library distinct from the Royal Library. According to his analysis, the Royal Library encountered too wide a public interest. 6 Students and professors had to have a place of their own, preserved from the intellectual appetite of the Berlin bourgeoisie. Also, it turned out that the students didn't respect the rules they had to observe when they borrowed books from the Royal Library -they even dared passing books to one another without telling the librarians, causing chaos and additional work. In order to allow the students to be able to consult all the necessary textbooks and topical scientific reviews, the University Library was to be a reference library. Wilken also negotiated with the Ministry on one side and with the Senate of the University on the other how high the budget would be, who would have to pay for it, how many rooms were necessary, what the assistant librarian would be in charge with. Wilken's aim was to create a small University Library that would allow the Royal Library to fulfill its missions better by being relieved of the task of helping out students.

The King did not consider the foundation of a University Library a really necessary step, but accepted to give his blessing as long as it was financially self-sufficient. The expenses for the purchases were to be carried on the one hand by the leftover of the heating and lighting money each student had to pay for, and on the other hand by a subsidy which every person receiving their doctor or professor title or being appointed at the University had to donate. And the University Library remained in the rooms of the Royal Library for ten more years, where it was allotted not the three rooms Wilken wished, but only two.7

The University Library was not granted a decent budget by the State until the late 19 th century. By that time, several scholars and teachers had increased the collection by donating their personal library. This is the case of Wilhelm von Humboldt's linguistics collection, but also Jakob and Wilhelm Grimm's, August Boeckh's, Theodor Schmalz', Gustav Parthey's, August Twesten's books, and some more. 8 This inheritance contributed to the greatness of the University Library in the early 20 th century.

Two directions of interpretation seem to me to be of interest concerning the development of the institution:

 One can observe that some major contributions were made by single individuals (the scholars who donated their private collections or whose collections were purchased) and not lead by the institution itself. This point is well illustrated by the example of Humboldt, the Grimms or Boeckh, thanks to whom the philological section was particularly well represented within the Humanities, and also contributed greatly to the scholarly reputation of Berlin in the late 19 th and early 20 th century.

 Neither King nor Minister considered it a priority to set strong ideological guidelines. For all of his poverty, Wilken had the freedom to acquire the books he considered necessary. The University Librarian was given the opportunity to, for instance, have a certain interpretation school better represented than the other. This is also true for the donations. Humboldt, Grimm, Boeckh did not only have their field of specialty, they also had ideological interests which lead to the over-or under-representation of parts of that field in their book collections. All in all, the University Library has been shaped by a handful of Librarians and Scholars -and generations of students were trained according to their choices. Interestingly enough, the freedom of teaching and learning, which is considered as one of Humboldt's greatest achievements and which was in reality very relative considering the many ways censorship was implemented, could express itself more fully in the constitution of the Library. It was poor, but free.

The foundation of the Chicago University Library

The direct continuity between the Berlin and the Chicago story is obvious. It begins in fact in Evanston with what Jeffrey Garrett, Director of Special Collections at Northwestern University, calls "a tale of two cities".9 In the 1860es, the Librarian of Northwestern University was able to acquire the books of Johannes Schulze, a passionate book collector who was also a major protagonist in the Prussian education system. Schulze contributed greatly to establishing Hegelianism as the dominant ideology at the Berlin University. 10 Northwestern University was able to buy his book collection that the Prussian State, by then in war with France and accordingly budget-wise pretty tight, had just turned down. This is not the only example of the Berlin-Chicago connection. Some thirty years later, still in Chicago, a handful of entrepreneurs dreamed of establishing a University of their own. They asked William Harper, a well-known Yale scholar, to be its founding President. The very first thing Harper did was -to constitute and acquire a scholarly rich University Library. He was hoping that this would "legitimatize the University's broad commitment to education as well as cast an aura of scholarly respectability over the whole enterprise." (Rosenthal, curator of the Special Collections) 11 So Harper set sail for Europe, and more precisely for Berlin, in 1891. "Within two weeks of his arrival in Berlin, Harper received from Simon an offer for the entire contents of the Calvary stock, which was stated as containing 300,000 volumes and 150,000 pamphlets. The large scale no doubt appealed to Harper's sense of enterprise, for never before had an American university acquired so many books at one stroke. He could also see the Calvary books as part of the strategy that would be necessary once he returned home. He (…) would (…) have to enter immediately into a fund-raising drive that would literally create a campus and get the University started within a year. Harper saw beyond the books to the attention which they would draw to the University. " 12 The negotiations proved harder and the deal not quite as good as it seemed in the first place. The seller didn't deliver as many volumes as he had originally promised. And although the board of trustees had secured the money for acquisition and shipping of the books, the hope for an adequate building was not really fulfilled according to Harper's wishes. It was only 1906 that the plans for a library building were realized -a building that would significantly carry Harper's name.

Rosenthal states: "But the collection did yield other results. The University Library became the largest in the city, and by 1896 it was the second-largest university collection in the United States with 340,000 volumes, the Berlin Collection contributing its uncertain but dominant share. This rapidly achieved status was used by Harper to lure faculty and also to give a sense of stability to the University." 13 This did not only concern the scholars. The press had largely contributed to the myth of the deal of the century and hence to establishing a little bit more what would be a long-lasting connection between business and scholarship in the US.

[Let's take a closer look at the content of the Berlin Collection. Classical Philology, Archaeology and Natural Science were the three fields which were particularly well represented. Simon highlighted the following aspects in his offer: "For "certain branches of classical philology and principally in classical archaeology the stock forms the richest existing library in the library world so that it is superior to the British Museum and the Royal Library in Berlin." (…). Simon (…) noted that the greater part of the manuscripts came from the library of Pope Pius VII and included such "treasures" as three autograph letters of Raphael, the original manuscript of Abbe Rance's work against Mabillon, an unpublished work of Friedrich von Schlegel, and a fourteenth-century Book of Hours "with unknown French poems illustrated by an artist of the early Burgundian school."14 Although several of these informations proved erroneous, this description sketches accurately what was considered as the greatest scholarly advantage of the collection -and also what was considered as an attractive business offer. Both aspects show Philology as a leading discipline in the late 19 th century.] Let's draw a small comparison between the Berlin history and the Chicago history.

 The foundation of the Berlin University was in many ways a patriotic act. Being able to teach and perpetuate scholarship after Prussian standards, being a scholarly active nation in spite of the war and its economic consequences, those were actual motivations for the founding of the University. In this setting, the University Library played a secondary role, mainly because such institutions as the Academy of Sciences, the Royal Library, etc. were already in place. Starting from scratch, the Chicago people inverted the process and put the University Library in the center of the University project as a way to make up for the lack of every other institution, or a core around which the other parts would grow by accretion. Instead of referring to a national tradition, they aimed at founding one by keeping allegiance to the world-wide recognized German philological tradition.

 The input of civil society in early 19 th -century Berlin was to be found on the level of intellectual exchange; in Chicago, it was of economic nature. The relationship between institutionalized scholarship and the non-Academic world then necessarily evolved differently in both countries.

I would now like to take a look at these institutions today and see what they tell us about what civil society expects from Academia.

Setting new standards: preservation, digitization and new buildings

Today's University Libraries face new challenges compared to their ancestors, but they face more precise ones. First, they are in charge of the preservation of books and manuscripts. The standards in that regard have evolved dramatically since the 19 th century. Librarian Science has developed preservation techniques and set preservation priorities. Not all media will be preserved in the same way, but the older media, partially because of their rarity, are an object of particular attention. Ultimately, the most precious documents will be engraved in miniature in nickel, a non-corrosive metal that would be likely to survive a global nuclear catastrophe and would only require the basic cultural skill of producing a magnifying lens in order for people or whatever form of intelligent beings to be able to retrieve information from it.

Second, University Libraries have to offer a space where to work on their collections. These missions are at the core of both brand new University Library buildings in Berlin and Chicago. The architectural choices are different, one building choosing the underground, the other one the elevation, both working with glass elements. The Reading Room is the core of Mansueto Library. If you need a book, you order it online, a robot picks it up for you and delivers it within 15 minutes in the spot of your choice. The work atmosphere is radically different from the tradition of thick walls and skylessness that usually prevails in Libraries. The Jakob-und-Wilhelm-Grimm-Zentrum made the choice of displaying the books for the users to pick them up. Both libraries encountered a success that actually disrupts their effective functionality. Groups of tourists thrive through Mansueto, while it is almost impossible to find a free seat at the Grimm-Zentrum. The Grimm-Zentrum, of course, has a more complex task towards non-University public than Mansueto, which is open for use to students and professors of the University of Chicago only.

Both examples show how the University Library has become a recognized space for scholarly work, not only inside the Academia, but also in a strong connection to civil society. The Grimm-Zentrum was not acquired with the students' heating and lighting money, but with the University's own budget. Mansueto carries the name of the generous Alumni who funded its construction. Not that the Grimm-Zentrum doesn't count on exterior donators; especially Alumni are invited to contribute by donating money to restore the Grimm's books. But the Library itself is not named after the donator,15 it is named after the intellectual inspiration and inheritance. It is a different choice, one that reconstructs a bridge to the age of the founding fathers. You can see it also a recognition of a past that doesn't require a direct confrontation with the third Reich, in a city that is architecturally marked at each corner by world war 2 and cold war.

Let me come to my last point concerning the mission of University Libraries today. Third and last, they have to grant access to their collections and keep those up to date. Many of the books that were representative of contemporary scholarship at the beginning of the 19 th century -not to speak of the manuscripts -have already begun to deteriorate. The best way to make those accessible -and for that matter, not only to Academics, but also to a wider audience -is to digitize them. I will not get too deep into the meaning of digitization for the world of Libraries in general and University Libraries in particular. More and more conferences are dedicated to the technical aspects connected to it. I will just state a few elements that can shed light on the questions of this conference.

A digitization is a very fragile medium, but it has the advantage of being susceptible of being saved on different servers, in different places, which increases the likeliness for it not to disappear. Also, it is a way of communicating knowledge without much infrastructure: once the digitization is online, you need neither a librarian nor, for that matter, a robot to get the book for you. Whereas the Mansueto Reading Room is not open to public, all of the books and manuscripts digitized by the UofC library are accessible online to anyone. But the digitizing process takes infrastructure. So the real important question is: What has been digitized? When a digitization is not determined by preservation reasons (if a book is falling apart for instance), it has to be part of a Library project -and projects are defined mostly by the money they bring in, but sometimes also by the way they relate to the history of the library.

The Digital Culture is in the process of changing radically the way we read texts and the way we give them to read to others. This change of paradigm affects philology in particular. The roles of editor and interpret are being redefined. Both have to be able to deal with hermeneutical as well as technical issues. The text layers between the establisher of the text and its recipient are multiplied. Many moments of encoding, decoding, organizing query systems in complex databases and implementing search engines, mediatize the relationship the scholar has to his text: it is not just your text editor and you. It is not just your 500 exemplars of your book. It is making a text accessible, potentially, to people you don't know of and who will be interested in your text for a reason you cannot imagine. Libraries are accountable for that change too and in that sense too, they relate directly to their 19th-century ancestors. University Libraries, by then as well as today, were forges for a new kind of philology, trying to figure out how to edit and interpret.

University Libraries and civil society -by way of a conclusion

Although they have established themselves as a part of Universities, University Libraries remain today in an ambivalent position. Their primary role is to serve the University they are part of, but they also remain a platform between Academic world and civil society. They occupy a singular space through which each nation relates not only -and maybe not mainly -to their political, but to their intellectual history. The dilemma which Wilken had to face is still present in Berlin: the University Library has a limited budget, a great scholarly mission and opportunities within which it has to make choices. Which of the major scholar's inheritance should be reconstructed first? Grimm, Boeckh, Schmalz? The dilemma Harper had to face is also still present in Chicago: The Library needs investors to be able to take care of the precious collections it is harboring. How far should the recognition towards the generous donators go?

In Berlin as well as in Chicago, institutional structures and collections did not come together as parts of a big plan, but accumulated partly randomly in the course of history, as a deal between reality and ideal that had to be perpetually re-negotiated along time. Interestingly, this historical construction has acquired the status of a funding myth in the institution, although the knowledge we and the University Library have of them is still blurry. To this day, there is no list of the books that originate from the Berlin Collection in Chicago, nor is there a list of the books, for that matter, that belonged to Boeckh in the Berlin University Library. The golden age of glorious philology is surrounded by a foggy aura. I am not saying that such idealizations of historical constructs are not necessary. They are of need, especially to create connections with a wider audience. But it is our task to allow this wider audience to get deeper in the reality of it. University Libraries are invaluable platforms when it gets to bringing students closer to research, when it gets to making civil society aware of fact that not everything has been said yet about the past, that the past constructs itself in the present and the future. It is a huge chance for a us that University Libraries have actually evolved to being what they now are, buildings that shape the city, collections that give it an intellectual imprint, and that acknowledge their history. Berlin will never be Chicago and Chicago will never be Berlin -but as long as they both digitize the history they have in common, we will be able to look into mysterious, old manuscripts and see how close they are to us.

Friese, S.

2 QUOTE ANNA.3 Friese, S. 4.

HIER REF ART. ANNA

Friese, S.

6-7.

Wilken quoted by Friese, S. 11.

Friese,

Vgl. http://www.konyvtar.elte.hu/hirek/rendezvenyek/2004/digit_konf/ebp_ea.htm

http://www.library.northwestern.edu/node/2618

http://www.library.northwestern.edu/node/2618

http://www.lib.uchicago.edu/e/spcl/excat/berlin/history.html

http://www.lib.uchicago.edu/e/spcl/excat/berlin/history.html

http://www.lib.uchicago.edu/e/spcl/excat/berlin/history.html

http://www.lib.uchicago.edu/e/spcl/excat/berlin/history.html

Add here ref to the donor-website listed on the webpage.


D L Gonzalez
email: gonzalez@bo.imm.cnr.it

S Giannerini
email: simone.giannerini@unibo.it

R Rosa

Circular codes revisited: a statistical approach

Keywords: Comma-free codes, Circular codes, Reading frame synchronization, Statistical analysis, Protein synthesis accuracy

discovered the existence of a common circular code in eukaryote and prokaryote genomes.

Since then, circular code theory has provoked great interest and underwent a rapid development. In this paper we discuss some theoretical issues related to the synchronization properties of coding sequences and circular codes with particular emphasis on the problem of retrieval and maintenance of the reading frame. Motivated by the theoretical discussion, we adopt a rigorous statistical approach in order to try to answer to different questions. First, we investigate the covering capability of the whole class of 216 self-complementary, C 3 maximal codes with respect to a large set of coding sequences. The results indicate that, on average, the code proposed by Arquès & Michel has the best covering capability but, still, there exists a great variability among sequences. Second, we focus on such code and explore the role played by the proportion of the bases by means of a hierarchy of permutation tests. The results show the existence of a sort of optimization mechanism such that

INTRODUCTION

All the steps of genetic information processing that characterizes normal functioning of living organisms share an astonishing level of accuracy that pervades all the biochemical processes involved. DNA replication, RNA transcription and mRNA editing, protein translation, represent outstanding examples of such capabilities. In particular, protein translation accuracy depends mainly on the synthesis process performed at the level of the active site A of the ribosome and the correct charging of tRNA's with their cognate amino acid. But if the ribosome complex is able to maintain high levels of translation accuracy, as is experimentally observed, the mRNA template has to carry sufficient information for ensuring a correct decoding behaviour.

Thus, the mRNA base sequence needs to include (explicitly or implicitly) appropriate information regarding: punctuation signs, direction of translation, synchronism with the normal reading frame and faithful assignation of codons to cognate amino acids. In fact, different organizational structures of mRNA sequences have been associated to these functions and the ability of the genetic code of carrying different levels of information along gene sequences has been recently remarked [START_REF] Itzkovitz | The genetic code is nearly optimal for allowing additional information within protein-coding sequences[END_REF].

In some cases, codon translation has been shown to be context depen-dent; this indicates that linear reading of codons does not suffice for ensuring faithful translation of amino acids and punctuation signs. It is well known that this is particularly true for start and stop signals for which the presence of specific sequences of bases are needed in order to confirm the function of the punctuation codons. This is the case, for example, of the Shine-Delgarno sequence situated some bases up-stream to the start codon site in prokaryotes. The need for sophisticated biochemical methods of mRNA information reading for translation accuracy has been put in terms of error correction of synthesis start signals [START_REF] May | Analysis of coding theory based models for initiating protein translation in prokaryotic organisms[END_REF]. Also, in order to ensure a correct amino acid translation different reading hypotheses have been proposed; these include the use of intron information [START_REF] Fordsyke | Are introns in-series error-detecting sequences?[END_REF] and non-linear decoding of di-nucleotides as a proof-reading mechanism (Gonzalez, 2008b,a).

Another important factor related to the accuracy of protein translation is the maintenance of the correct reading frame. This capability of the translation machinery has been associated to base periodicity and to the existence of circular codes in coding sequences. In particular, base periodicity has been identified on the purine/pyrimidine alphabet at the gene and gene population levels by (Shepherd, 1981;[START_REF] Fickett | Recognition of protein coding regions in DNA sequences[END_REF]Michel, 1986). After these pioneering works, several researchers, studied these results on the other 2-letter alphabets (strong/weak, etc.) and the 4-letter alphabet [START_REF] Arquès | A complementary circular code in the protein coding genes[END_REF]. Normal reading frame synchronism has been related to the presence of circular codes in the structure of protein coding sequences [START_REF] Michel | A relation between trinucleotide comma-free codes and trinucleotide circular codes[END_REF]Trifonov, 1987). Such codes also explain the purine/pyrimidine periodicity [START_REF] Arquès | A complementary circular code in the protein coding genes[END_REF].

In [START_REF] Farabough | How translational accuracy influences reading frame maintenance[END_REF] it is shown experimentally that the ac-curacy in amino acid translation is related to the accuracy of frame maintenance. This fact is remarkable because it indicates a complex interaction between the two informational levels similarly to what happens in error detection/correction codes developed for transmitting digital information in technological applications.

In this paper we study the organizational structure of mRNA information in relation to the property of frame detection and maintenance by using circular codes properties. In section 2 we present a theoretical discussion on code theory in theoretical biology with particular emphasis on comma-free codes and on circular codes. The theoretical issues arisen from the discussion are studied in Section 3 by adopting a rigorous statistical approach. In particular, we aim at i) quantifying the fitness of an arbitrary circular code with respect to a single sequence or an ensemble of coding sequences ii) investigating the existence of some sort of optimization mechanism that links circular codes to coding sequences. These task are accomplished by setting up a series of tests based on appropriate resampling techniques. In Section 4 we discuss the results and outline the conclusions.

The main results point to the existence of an optimization mechanism such that coding sequences maximize or minimize the coverage of circular codes on specific reading frames. Also, the two levels of information regarding amino acid recognition and frame maintenance, are indeed independent. This means that a coding sequence can carry the information ensuring frame synchronization independently from the information regarding accurate amino acid translation. Of course, the synthesis mechanism, actuated by the ribosome, can take profit simultaneously of both kinds of information in a synergic process for ensuring overall protein translation accuracy. The results also shows that the roles of bases according to their relative position along codons are differentiated: frame maintenance seems to be mainly associated with the first two bases, while amino acid translation accuracy is mainly related to the third base.

SYNCHRONIZABLE CODES

Comma-free codes

A typical problem encountered in the transmission of a message composed by words made up of strings of symbols taken from an alphabet is that of synchronization. In order to ensure a faithful decoding of words we need not only a faithful decoding of symbols but also an accurate signalling and decoding of start and end points of words. This allows to group the alphabet signs as to form admissible words. In common written language this last task is implemented by using a separation sign (the blank character) between adjacent words. For example the following message: SOME ANTS ARE MIGRANTS is easily interpreted by identifying any word of the message as the string between successive blanks. Now, if we eliminate the blanks the message reads, SOMEANTSAREMIGRANTS Now, the semantic interpretation of the message is ambiguous because it can be decoded in at least another different way, i.e.,

SO MEAN TSAR EMIGRANTS

All the words decoded in this last way are valid words of the English language and all the alphabet symbols have been correctly decoded; still, this alternative decoding is erroneous. The bug resides in the identification of start and end points of the words, indeed, a frame decoding error. The blank symbol is used for separating adjacent words but any other symbol not used in word composition, that is, other than letters of the alphabet, can do the job as well; since commas are used for separating parts of a sentence, comma is a symbol that is naturally used for the former scope; for this reason, a code that can be decoded without the need of separation symbols is said a comma free code.

Moreover, if in a digital communication system all the transmitted words have the same length (as is the case, for example, of bytes of fixed length in binary computer systems), the problem of word separation becomes a problem of synchronization. Since the decoding operation in binary messages is implemented at a constant symbol rate, say t symbols/second, the time for decoding words of constant length n is constant and equal to the time needed to decode n symbols, i.e., nt seconds. Thus, a correct decoding of words is achieved by starting the decoding process exactly in synchrony with the start of a word and resetting for decoding a new word periodically after nt seconds.

As the inverse of a time should be interpreted as a frequency, the problem is equivalent to keeping at the same time the word decoding frequency at 1/(nt) and the symbol decoding frequency at 1/t. Now, maintaining two frequencies in a fixed rational ratio, i.e., (1/nt) /(1/t) = 1/n, represents a problem of frequency synchronization; in this spirit, we call the general problem of word frame decoding, a synchronization problem. Frequency synchronization allows to achieve accurately the word length; of course, we need also phase accuracy for localizing correctly the starting points and reading the words in their correct frame. Consider the following example: THE OWL WAS HOT (correct reading frame) T HEO WLW ASH OT (frame-shift +1 reading)

In this example we see a sequence of words of three letters. Thus, the word decoding frequency needs to be 1/3 of the symbol decoding frequency. By ensuring a frequency synchronization at this value (1/3) we achieve a correct length reading of words. But if we use a wrong starting point (phase shift) the words are decoded incorrectly. This is shown in the second line of the example where a frame-shift of 1 symbol (+1 frame-shift) has been introduced. Observe that now most of the words have no sense so that we can argue that an error has been introduced. This is the main idea behind comma free codes: a frame-shift reading produces non-allowed words.

If the synchronization is ensured only at the beginning of one specific message, any perturbation along the decoding process can break the synchronization and, thus, the correct decoding of the reading frame; for this reasons, man made digital communication systems implement means for monitoring (continuously or intermittently) the quality of frequency and phase synchronization and possibly restore them in case of errors. Coding regions of DNA are very similar to digital computer messages where the alphabet symbols are the four bases (A,T,C,G) and individual words are formed by strings of three symbols forming an mRNA codon. The meaning of the words in coding sequences of DNA correspond to the coded amino acids which are sequentially assembled as to form the proteins in the ribosome complex; this is the so called translation or protein synthesis step in the central dogma of molecular biology. Remarkably, the decoding frequency of this process can attain more than 20 amino acids per second in E. coli at 37 • C [START_REF] Bremer | Feedback control of ribosome function in escherichia coli[END_REF]. As in the transmission of any digital message, a faithful protein synthesis needs appropriate means for ensuring synchronization of the decoding process with the correct reading frame of codons. Such ability is called reading frame maintenance; in this context frequency synchronization is not important because a correct word length reading is ensured by the stereo-chemical properties of the ribosome; instead, the correctness of the starting point is fundamental because once a frame-shift decoding is started it continues producing wrong amino acids until a wrong (out-of-frame) stop codon is encountered and the synthesis is stopped. A loss of frame maintenance (or frame-shift) produces a completely erroneous translation because the genetic code is compact (any sequence of three bases represent an aminoacid) and frame-shift reading errors usually lead to non-equivalent codons, that is, codons that represent different amino acids (as it can be deduced by inspection of the genetic code).

The problem of synchronization (or reading frame location and maintenance) in mRNA translation has been identified from the beginning of molecular genetics. [START_REF] Crick | Codes without commas[END_REF] proposed an ingenious solution for the problems of both amino acid coding through codons and frame maintenance. They proposed an ancestral genetic code with the property of being comma-free and non degenerate. In doing so, [START_REF] Crick | Codes without commas[END_REF] introduced the comma free codes for the first time, an interesting example of communi-cation theory problems generated by a biological insight. Indeed, these codes have provoked a great interest from the mathematical point of view of coding theory (see e.g. [START_REF] Golomb | Comma-free codes[END_REF]. The main idea of comma-free codes is to use a subset of available words in such a way that a concatenation of valid words in the correct reading frame produces inadmissible words when it is read out of frame. In this sense, comma free codes are error-correcting codes. A classic example is represented by the comma free code implemented in a two symbol alphabet, i.e., the R,Y alphabet (R=Purine=A,G; Y=Pyrimidine=T,C). This alphabet has been proposed as a primeval alphabet related to the origin of the genetic code. If we consider the set formed by the two codons {RRY, RYY} they can be concatenated in the following ways: RRYRRY RRYRYY RYYRYY RYYRRY If we allow any possible reading frame, the newly generated codons are: RYR, YRR, YRY, and YYR. Notice that all of them are not allowed codons, hence, the original set, {RRY, RYY} represents a comma-free code for words of three-letters over an alphabet of the two symbols, R,Y. If this code is used for coding amino acids, there are two valid words out of 8 (2 3 ) that have the property of automatic frame retrieval (there are no valid words in the out of frame reading). However, the price to be paid is the high redundancy which amounts to 3/4 of the total quantity of information (2/8 = 1/4 is the proportion of valid words). Furthermore, despite the great interest arisen, these kind of codes have not been found in current mRNA protein coding sequences but could have existed in primitive genes on a purine/pyrimidine alphabet.

Circular codes

Another class of codes that allows frame retrieving with less redundancy, is represented by the so called circular codes. These codes have the property of synchronizability, that is, they allow to retrieve the correct reading frame by using an appropriate window of mRNA bases. Circular codes obey less restrictive rules than comma-free ones. As shown before, comma-free codes are based on a kind of zip coding; only some codons have a sense and these are always placed in the correct reading frame. In fact, when such codes are built on three letter words over an alphabet of four they should have at most 20 meaningful codons out of the 64 possible. Circular codes, instead, are characterized by less redundancy. They possess the circular property, i.e., any word written on a circle (the last letter becoming the first in a torus like fashion) can be decomposed in at most 1 way in words of the circular code.

Suppose that a given codon belongs to a specific circular code; due to the circular property, the same code cannot contain any circular permutation of this codon. This is because the concatenation of a given codon with itself generates all its circular permutations if read in an out-of-frame situation.

For instance, if the codon ATC belongs to a given code then its circular permutations -TCA, and CAT, which can be generated by reading out-of-frame the concatenation of the codon with itself ATCATCATC -need to be ex-cluded from such code. Now, since the identical codons, AAA, TTT, CCC, GGG, coincide with their circular permutations, these codons are immediately excluded from any circular code. If we eliminate these codons from the 64 possible ones, we are left with 60 codons that can be grouped in 20 sets of three codons each. These sets are built by a codon together with its two circular permutations. In this way we can form an arbitrary circular code by choosing a number of codons from the 20 sets.

A circular code that contains exactly 20 codons is called a maximal circular code. The total quantity of maximal circular codes is 3 20 . Thus, the probability of generating by chance a particular maximal circular code is 3 -20 ≈ 2.9 × 10 -10 . For this reason, it is remarkable that specific maximal circular codes (and also more restrictive versions of them) have been actually found in genes. In fact, by studying the probability of occurrence of codons in the reading frames [START_REF] Arquès | A complementary circular code in the protein coding genes[END_REF] have identified a particular maximal circular code. The codons that form such code, called the X0 code, have a preferential frequency of occurrence in the normal reading frame; moreover, the authors assert that X0 is a common code for both Eukaryotes and Prokaryote organisms. The code X0 exhibits many interesting symmetry properties; for example, the first circular permutation of X0, called X1, is found preferentially in the +1 out-of-frame condition; also, the second circular permutation, X2, is found in the +2 (or-1) out-of-frame condition. X1 and X2 are also maximal circular codes. This property of X0 is called the C 3 property. Moreover, the X0 code has also a peculiar characteristic regarding its symmetry under the complementary transformation, i.e., when the bases of a codon are changed by their complementary ones (A↔T, and C↔G), and the codon is read in the reverse direction. The X0 code is invariant under this transformation, or, in other words, the X0 code is self-complementary.

Other maximal circular codes have been found in different life domains (bacteria, archea, and mitochondria) (Arquès and Michel, 1997;Frey andMichel, 2003, 2006), but these codes do not have the self-complementary property.

However, no biochemical mechanism linked to the functional use of circular codes has been reported so far. For this reason, tentative explanations for the existence of circular codes have been proposed. For example, [START_REF] Koch | About a symmetry of the genetic code[END_REF] noted that the distribution of bases occurrence in the different codon positions can generate in a natural way some circular codes. Assuming that these frequencies are related by a self-complementary symmetry, the generated codes share also such property. From these assumptions, as shown by [START_REF] Lacan | Analysis of a circular code model[END_REF], it is possible to generate 88 maximal, C 3 and self-complementary circular codes. However, in the same article the authors proved that the common Eukaryote-Prokaryote X0 code cannot be generated in this way.

The authors implement an algorithm (the flower automaton) for generating all the 216 maximal self-complementary circular codes. Since the proportion of bases in actual genomic sequences do not satisfy exactly the self-complementary condition the issue on which codes can be obtained by relaxing the conditions established in [START_REF] Koch | About a symmetry of the genetic code[END_REF] remains open.

Moreover, recent works (Arquès and Michel, 1997;Frey andMichel, 2003, 2006) reveal the presence of many other circular codes in different organisms; also, the existence of evolutionary mechanisms responsible of codon variations from the archetypical common X0 circular code are hypothesized [START_REF] Arquès | An evolutionary analytical model of a complementary circular code[END_REF][START_REF] Ahmed | Essential molecular functions associated with the circular code evolution[END_REF]. Moreover, [START_REF] Michel | A relation between trinucleotide comma-free codes and trinucleotide circular codes[END_REF] identify a relation between the comma-free codes and the circular codes by constructing a hierarchy of these two classes of codes. All these results prompt two fundamental questions: i) quantifying the fitness of an arbitrary circular code with respect to a single sequence or an ensemble of coding sequences ii) investigating the existence of some sort of optimization mechanism that links circular codes to coding sequences. In the next section, we explore these issues by means of a rigorous statistical approach.

STATISTICAL ANALYSIS

In their approach [START_REF] Arquès | A complementary circular code in the protein coding genes[END_REF] (see also [START_REF] Michel | A relation between trinucleotide comma-free codes and trinucleotide circular codes[END_REF]) start from a set of coding sequences and derive circular codes from the frequency distribution of the codons in the three reading frames of such set. In particular, they find a common circular code, (AM code in the following) from a large set of Eukaryote and Prokaryote coding sequences. Our approach is complementary to theirs in that we look for the best codes among the whole class of 216 C 3 codes. To this aim, we define a measure (called covering capability) for describing the ability of a given circular code in covering a mRNA sequence. Then, we study the covering capability of the whole class of maximal, C 3 , and self-complementary codes (denoted by C, in the following) in a sample of coding sequences. Moreover, we explore the covering capability of the 216 codes as a function of their distance from the AM code. We focus on the best code in terms of mean covering capability and assess the significance of the result by means of a bootstrap test. In addition, we analyse how the covering capability of a code is linked to i) the organizational structure of a sequence ii) the proportion of bases in the three codon positions iii) the reading frames. To this aim we implement a series of permutation tests.

Finally, we analyse in some detail the distribution of the covering capability of some of the best codes. The 216 C 3 codes used have been taken from the lists in [START_REF] Michel | A relation between trinucleotide comma-free codes and trinucleotide circular codes[END_REF].

I(a, A) = ⎧ ⎪ ⎨ ⎪ ⎩ a if a ∈ A ∅ else (2)
Notice also that xr, with r = 0, 1, 2, means x0, x1, x2.

As an example, let s = (c 1 , . . . , c 10 ) = ATGGATGCAGTTAAGAGTGAGGTTGCTCTG be a sequence composed by 30 bases, see Fig. 1. Denote the sequence read in frame as s0. Now take a (non-maximal) circular code z of 10 codons.

In the figure we report z0, i.e. the code z itself, z1, i.e., the first circular permutation of z0, and z2, i.e., the second circular permutation of z0. We see that out of the 10 codons of s0, there are 4 codons, GAT, CTG, GTT (twice), which belong also to z0 (in black in the figure); hence, we say that z0 "covers" 12 bases of s0. The same procedure is repeated with the code z1

as to obtain the covered bases of s1. In this case, the number of the covered bases of s1 is still 12. Lastly, we consider z2 and s2 and get 6 covered bases.

Taking into account the "union" of the coverages by z0, z1 and z2, we obtain the total covered bases of the sequence s, represented in black in the last line mean we call the average covering capability denoted by C x and defined as: whose average covering capability is around 66%. Moreover, the average covering capability ranges from ≈ 35.53% up to ≈ 66.88% that corresponds to the AM code. This result shows that also from this perspective, the AM code is a special one, in that, on average, it performs better than the other codes.

C x = 1 3248 3248 s=1 C x,s
We can go further into the matter by computing C x as a function of the set distance between each C code and the AM code. If x and y belong to the class C, then the distance δ x,y between them is defined as δ x,y = 20 -card{x ∩ y} (see also [START_REF] Ahmed | Essential molecular functions associated with the circular code evolution[END_REF]. The result is reported in Fig. 3.

The Figure shows that codes having average covering capability similar to the AM code are also close to it from the point of view of the distance δ AM,y . Note that the set distance δ AM,y is always a multiple of 2 as the codes are self-complementary. In particular, there are 6 codes at distance 2 from the AM code, that is, these codes differ from the AM code by 2 codons. Moreover, the greatest number of codes (42) are at distance 14 and at the greatest possible distance 20 there are 8 codes. As mentioned before, C AM = 66.88%. Notice that there are 11 codes with average covering capabilities greater than 65% (see dashed line in Fig. 3). Also, in the figure we have framed the points belonging to the subset of Koch and Lehmann codes (KL codes, in the following), formed by 88 codes taken from Table 3 of [START_REF] Lacan | Analysis of a circular code model[END_REF]. We see that 2 out of such 88 codes are at distance 2 from the AM code, and there are 3 codes with average covering capabilities greater than 65%, i.e. very close to the AM code. Notice that, for any given distance, the best code is of the non-KL type; on the contrary, the worst one is always of KL type. We can make a coarse assessment of the probability of the joint occurrence that by chance the best code is of non-KL type for all the distances. At a given distance 2k with k = 1, . . . , 10 we have that the event the best code is of KL type can be modelled as a Bernoulli variable with parameter p 2k = n * 2k /n 2k where n 2k is the overall number of codes at distance 2k and n * 2k is the number of codes of KL type at distance 2k. Hence, the joint probability is given by 10 k=1 (1p 2k ) = 0.0035. This result assumes that the variables are independent and that the codes are equally likely; still, we can be quite confident of the unlikeliness that such an occurrence happened by chance.

Consider now the circular code having the average covering capability equal to 66.33%, which is the second highest after the AM code (call it c2 ).

Figure 4 shows the distributions of the covering capability of such two codes.

From the figure, we see that, for instance, the AM code, around its maximum, covers 598 sequences out of 3248 (i.e. 18.4%) with a covering capability between 62.5% and 67.5%. It appears that both distributions are quite similar and bimodal in shape. The histograms are not clearly separated, so at first sight, we are induced to think that the difference between the average covering capabilities of the AM code and the one of c2, i.e., d = 66.88 -66.33 = 0.55%, is not significant. Nevertheless, by means of a rigorous statistical analysis, it turned out to be the opposite. Formally, the problem reduces to testing the null hypothesis H 0 , that the populations from which the observed samples have been drawn have the same mean or, equivalently, whether the difference of the mean values is zero. In this context, there are two aspects which deserve some care. The first one is that we are not legitimate to introduce the hypothesis of either normality or homoscedasticity, on which "traditional" statistical methods rest. The second aspect is that the samples under study are not independent; in other words we are in presence of paired data. A well-known example of such a situation is that of measuring on the same subject the effect (if any) of two different pharmacological treatments. If both measurements are made on the same subject, the variability between subjects is eliminated from the comparison, so that small treatment effects can be detected, even though the response of different subjects may be quite different. In our case, the "subjects" are the coding sequences and the "treatments" are the circular codes.

Because of the above mentioned issues, we exploited a nonparametric bootstrap method [START_REF] Efron | An Introduction to the Bootstrap[END_REF] for paired data. The procedure consists of the following steps. We test the null hypothesis H 0 : In the present case, the result of the bootstrap test is unambiguous: the bootstrap p-value is 0 so that the null hypothesis is rejected, that is, the average covering capabilities of the AM code is greater than those of other codes. Notice that, both the AM code and the c2 code do not belong to the KL subset. We have repeated the above analysis for the best code of such subset and have found similar results.

C AM = C c2 against H 1 : C AM > C c2 ,
In other words, also this latter code has i) a bimodal coverage distribution similar to the two codes studied above; ii) an average covering capability significantly lower that of the AM code.

In the following, we investigate in detail how the covering capability of the AM code is related to the structural information of a sequence under different randomization schemes (corresponding to different biological hypotheses) and for different frames. Contrarily to the coverage analysis performed above, this implies the study of the coverage of the AM code in terms of codons in the three reading frames, separately. In practice, we implement a series of permutation tests. The first scheme is produced by permuting without constraints the bases of the sequence (we denote this scheme by no constr.). In this scheme, for example, a letter from the first base position can be swapped with letters from the second or third base position; thus, only the global proportions of bases are preserved. Moreover, we explore a set of constrained permutation schemes that are associated to a hierarchy of hypotheses. In particular, we introduce schemes that involve the permutation of the bases in their position. In such a way, permuted sequences will have the same proportion of bases as the original sequence in the corresponding position. For instance, in the scheme denoted by (b 1 , •, b 3 ) the first and the third bases of each codon are permuted (preserving their position) while the second bases do not vary. The most structured situation is created by permuting only the third bases of the sequence (•, •, b 3 ). In this scheme, besides the proportion of bases, also the amino acid sequence is mostly preserved. Notice that the schemes (b 1 , b 2 , b 3 ) and no constr. are different in that in the former, letters are permuted by keeping their codon position. The permutation framework is similar to the bootstrap framework as the results of the two tests are asymptotically equivalent. In this context, however, permutation tests are preferred as they allow to preserve the proportion of bases in permuted sequences so that the hypotheses tested are biologically meaningful.

The main motivation of the inquiry is the following: there is a mechanism that allows the synchronization of the frame by means of circular codes; such mechanism is related to the covering capability of the codes. Now, the hypothesis we are testing is Does the mechanism depend on the proportion of bases in the three positions of the codons? Operationally, this translates into the following system of hypotheses:

As far as the covering is concerned:

⎧ ⎪ ⎨ ⎪ ⎩
H 0 : the original sequence does not differ from permuted sequences H 1 : the original sequence does differ from permuted sequences In other words, if H 0 is true if we permute the sequence and preserve the proportion of bases then the coverage obtained from the permuted series will be the same of that derived from the original series. The procedure is the following: for each sequence and for the AM code X0 together with its circular permutations X1 and X2 i) choose a permutation scheme and generate B randomly permuted sequences (e.g. B = 1000); ii) compute the coverage of the AM code on the B permutations for the three frames as to obtain the distribution of the coverage on the permutations. iii) perform a right tail test: reject H 0 if the coverage of the AM code on the original sequence is greater than the 95-th percentile of the permutations distribution.

The results are reported in Tab. 2 where we show the percentage of rejections of the tests over the 3248 sequences for each permutation scheme (rows), for the three frames and for X0, X1, X2. For instance, consider the scheme (b 1 , •, •) (only the first bases of the codons in the sequence are permuted); we see that for the sequences in frame and for X0, we reject H 0 in the 44.2% of the 3248 sequences; in other words, for 1436 sequences the coverage of the X0 code of the sequences read in frame is significantly greater than those of the permutations. The most striking results of Tab. 2 are the following. If we permute without constraints on the base position (no constr. scheme) we reject H 0 in about 80% of the sequences when the coverage of X0 (X1, X2, respectively) is computed upon in-frame (frame shift +1 and +2, respectively) sequences.

We refer to the combinations: code X0 -sequence in frame, code X1sequence frame shift +1, code X2 -sequence frame shift +2, as the natural combinations. This means that the coverage of the AM code in the natural combinations drops significantly when the sequences are permuted. This finding suggests the presence of some sort of informational structure of the sequences that the circular codes are able to capture. The next step is to test the hypothesis that such structure is related to the proportion of bases in their codon positions. This hypothesis is tested by means of the scheme (b 1 , b 2 , b 3 ). Clearly, the rejection percentage passes from 81.2 to 42.2 for the coverage of the X0 code on the sequences read in frame. Interestingly, the percentages drop from 80% to about 20% for the other two cases. Also, the permutation of just one base is sufficient to cause the drop. Apart from the natural combinations, the general trend indicates that, in practice, the coverage of the codes on the original sequences is never greater than those of permuted sequences in almost every scheme.

These results suggest that coding sequences might be optimized in a way as to maximize the coverage in the natural positions. On the other hand, with the previous analysis, nothing can be said on a possible minimization process acting out of the natural positions. In other words, consider a coding sequence read in frame; if such sequence is built as to maximize the number of codons that belong to the code X0, then, is it also built in order to minimize the number of codons that belong to X1 and X2? In order to answer to this question we need to implement the tests on the left tail of the permutation distributions. The procedure is the same of that described above with the exception of point iii) that becomes iii ) perform a left tail test: reject H 0 if the coverage of the AM code on the original sequence is smaller than the 5-th percentile of the permutations distribution. The results are reported in Tab. 3. For instance, consider the scheme (b 1 , •, •) we see that for the respectively. In conclusion, the results suggest that the proportion of bases in their codon positions is strictly connected to the organizational structure of a sequence and can play a role in the frame synchronization process. In the next section we review and discuss all the results in view of this perspective.

DISCUSSION AND CONCLUSIONS

In this paper we have analysed the capability of C 3 (AM-like) codes in describing circular properties of mRNA and DNA protein coding sequences.

The work of AM identifies an unique code for Prokaryote and Eukaryote coding sequences possessing particular symmetry properties. In particular the AM code is self-complementary, that is, if a codon belongs to X0 then also its complementary reversed version belong to X0. Moreover, the circular permutations of X0, that is, X1 and X2, are also maximal circular codes (though not self-complementary). In relation with symmetry properties, [START_REF] Koch | About a symmetry of the genetic code[END_REF] proposed that the AM code should be a consequence of a self-complementary relation regarding the frequency of bases in the different codon positions (along the normal reading frame). The KL model is based on the hypothesis of absence of correlation between successive bases in (protein) genes. Entropy methods, in particular, showed that this hypothesis is not verifed in current genes (Lacan and Michel, 2001, e.g.).

Therefore, the observation of a KL code in genes might be incompatible with a frequency dependence of bases in codon positions. In the framework of coding theory [START_REF] Lacan | Analysis of a circular code model[END_REF] showed that i) all the 216 codes can be generated with a flower automaton algorithm and ii) the AM code, as expected, does not belong to the KL class. However, it is not clear if this dif-ference in code building is significant at the functional biological level. The matter is important because, if circular codes can be generated simply from base proportions, then no particular organizational structure that relates a sequence to the circular codes is needed. We have investigated the issue from two different perspectives. In the first approach we have analyzed the covering capability of the entire class of 216 C 3 codes. Also we have compared systematically the covering capability of KL versus non-KL codes. In the second approach we have studied how the covering capability of a circular code is related to the dependence structure and to the proportion of bases of a coding sequence. This is achieved by destroying the dependence of a sequence in a controlled fashion by means of different permutation schemes.

Such schemes allow to preserve or destroy local and global proportions of bases as to build proper statistical tests.

Our first result confirm the primacy of the AM code over the entire class of 216 C 3 codes. This has been substantiated by means of a bootstrap test.

However, by using a set distance we found codes of the KL type that are very close to the AM code. Figure 3 shows the average covering capability of the 216 C 3 codes as a function of the distance from the AM code. Here, KL codes have been identified with a square. A simple visual inspection shows that the KL codes are interspersed over the entire set. However, for any given distance, the best code is of the non-KL type; on the contrary, the worst one is always of KL type. The probability that by chance the best code if of non-KL type for all the distances is about 0.0035.

Even if the average covering capability of the AM code is significantly higher than those those of other codes a look at the distribution of the cover-ing over the set of sequences shows an interesting scenario. In fact, the AM code, the second one in average covering capability, and the best code of KL type, have similar distributions (see also Fig. 4). All of them are bimodal with maxima placed around the same values, and all of them show similar dispersion features. The dispersion of the distribution becomes an important aspect of the discussion because it is intimately related to the validity of the original hypothesis of the existence of a single common code. In fact, the high variability of the covering capability, which can be as low as 35.5% for the AM code, shows that some codes (including KL ones) perform better than the AM code for specific sequences and/or classes of proteins.

In the second part of the analysis we have studied the connections between the organizational structure of a sequence, the covering capability of the AM code, the proportion of bases and the reading frames. To this aim, we have implemented a permutation test framework. The results show that there is always a preferred frame for the AM code and its circular permutations. A permutation that preserves the global proportions destroys the covering in nearly 80% of the sequences. But what happens with the remaining 20%? Probably, due to the high dispersion discussed above, the left tail sequences are those that the AM code cannot cover well. Thus, the coverage is not destroyed because there is not coverage at all. If we implement a similar permutation on the base positions but keep the base frequencies (scheme (b 1 , b 2 , b 3 )) we obtain a drastic fall in the percentages (from 80 to approximately 45%). This means that nearly half of the sequences lose their circular properties also when the frequency distribution of the bases in their position is maintained. This fact represents a clear evidence of the existence of an organizational level beyond the frequency distribution; still, it seems that this property is not universal as it is present in only one half of the sequences. In fact in approximately the 35% of the sequences preserving the local proportions of bases is sufficient for maintaining the circular properties.

Of course, this analysis is put forward for the AM code and does not ensure that other codes do not give different results. Again, this fact points to the need of defining in biological terms the appropriateness of a given code for describing a specific sequence.

Another interesting result is obtained by permuting only one letter at a time. It can be observed that the most stable situation is represented by permuting the third letters. This means that the circular coding is most insensible to permutations in the third letter. Since the third letter represents the true degree of freedom of the genetic code, it seems that synonymous codons can be chosen relatively freely without destroying the circular coding features of the entire sequence. In this way, the third letter might be used for implementing error correction mechanisms that do not interfere with the frame synchronization process.

In conclusion, the analysis presented contributes to shed light upon the role played by circular codes in the processing of genetic information. At the same time, new questions arise and more theoretical work is required.

A direction of future investigation is that of exploring the connections between circular codes and the theory associated to the mathematical model for the genetic code presented in [START_REF] Gonzalez | Can the genetic code be mathematically described?[END_REF]Gonzalez ( , 2008b)); [START_REF] Gonzalez | Detecting structure in parity binary sequences: Error correction and detection in DNA[END_REF][START_REF] Gonzalez | Strong short-range correlations and dichotomic codon classes in coding DNA sequences[END_REF][START_REF] Gonzalez | The mathematical structure of the genetic code: a tool for inquiring on the origin of life[END_REF]. Preliminary studies seem to indicate a clear relation between circular codes, global transformations and dichotomic classes.

  We define the covering capability C x,s of a circular code x ∈ C for a sequence s, a parameter which describes the number of bases covered simultaneously by the code x0 in frame, x1 on the frame-shift +1 and x2 on the frame-shift +2 conditions. Formally we have the following: Definition 1. Let s = (c 1 , . . . , c n ) be a sequence of codons where (s0, s1, s2) denote the sequence s read in frame, with frame shift +1, with frame shift +2, respectively. In terms of bases, the i-th codon of the r-th frame shift is denoted by {c r i } = (b 3i+r-2 , b 3i+r-1 , b 3i+r ). Define the coverage of the circular code x = {x0, x1, x2} over s as: in the following, card [A] is the cardinality of the set A. I(a, A) is the function defined as:

Figure 1 :

 1 Figure 1: Scheme explaining the covering capability C z,s of a circular code z formed by 10 codons, with respect to a sequence of codons s. The circular code is represented as z0, while z1 is the first circular permutation of z0, and z2 is the second circular permutation of z0. s0, s1 and s2 denote the sequence s read in frame, with frame shift +1, with frame shift +2. The black letters in C z,s are the bases of s covered simultaneusly by z0, z1, z2 circular codes. The dashed lines guide the eye between codons.

of

  Fig. 1, namely 20 bases. Finally, the covering capability (in percentage) results C z,s = 20/30 × 100 = 66.7%. In order to study the covering capability of the whole C class, we use a data set consisting of 3408 nucleotide sequences obtained from genbank 1 by means of the R package seqinr 2 (Charif and Lobry, 2007). In detail, we have extracted all the coding sequences from the 13 classes of proteins reported in Tab. 1, where the number of sequences analyzed for each class is listed. Such ensemble has been reduced by eliminating duplicate and short (< 120 base pairs) sequences. The final data set consists of 3248 sequences. For each code x ∈ C we have obtained a distribution of covering capability whose

Figure 2

 2 Figure 2 displays the histogram of the average covering capability C x of the 216 circular codes. The dashed line represents a kernel density estimate. The distribution appears bimodal, with one peak around 53.8% and a second one around 63.8%. Such bimodality is representative of a cluster of 12 codes

Figure 3 :

 3 Figure 3: Distance from the Arquès and Michel code versus average covering capability of each circular code. The 88 points framed with a square belong to the subset of Koch and Lehmann codes.

Figure 4 :

 4 Figure 4: Distributions of the covering capability of the Arquès and Michel code (continuous line) and of a code c2 ∈ C with the second highest average covering capability (dashed line).

  where C AM and C c2 denote the population means. Let {d s } = {C AM,s -C c2 ,s }, s = 1, . . . , 3248, be the observed sample of the differences with mean d = C AM -C c2 = 66.88 -66.33 = 0.55%. A bootstrap sample is obtained by randomly sampling 3248 times, with replacement, from {d s }, this means to bootstrap the values C AM,s and C c2 ,s (s = 1, . . . , 3248) in pairs. As a result, any d s can be drawn more than once, or not at all. Let {d * s } 1 the first bootstrap sample, that is the first sample formed by the resampled values from {d s }. Compute the first bootstrap replication d * 1 , as the mean of {d * s } 1 . Repeat the above computation a large number of times B (e.g., B = 1000) to obtain B bootstrap replications d * b , (b = 1, . . . , B). As usual, we choose the level α such as 0.05 or 0.01 and reject H 0 if the proportion of the bootstrap replications less than 0 (i.e. the bootstrap pvalue card{ d * b < 0}/B) is lower than α.

Table 1 :

 1 Classes of proteins whose sequences have been analysed. The third and forth columns report the number of sequences and the number of kilobases (kb) of each class,

	respectively.
	protein	No. of seqs	kb
	1 albumin	142	101.5
	2 alpha-globin	57	15.2
	3 beta-globin	141	50.3
	4 carboxypeptidase A	31	35.2
	5 globulin	6	2.9
	6 glycogen synthase	346	458.3
	7 heat shock protein 70	1022	1148.1
	8 insulin	45	9.9
	9 lactate dehydrogenase	164	113.7
	10 lysozyme	327	161.5
	11 phosphoglycerate kinase	1077	1221.7
	12 phosphorylase kinase	10	18.2
	13 troponin C	40	17.6

Figure 2: Histogram of the average covering capability C x of the 216 circular codes. The dashed line shows a gaussian kernel density estimate.

Table 2 :

 2 Percentages of rejections of the permutation tests over the 3248 sequences at level 95%, right tail, for the three frames and for X0, X1, X2. Each row corresponds to a different permutation scheme.

			in frame		frame shift +1	frame shift +2
	bases	X0	X1	X2	X0	X1	X2	X0	X1	X2
	(b 1 , •, •)	44.2 11.5 3.4 13.2 12.1 21.1 15.3 15.3 11.0
	(•, b 2 , •)	39.4 7.9 6.6 17.8 18.0 17.6 9.0 4.5 29.2
	(•, •, b 3 )	34.8 11.8 10.2 9.8 15.8 10.5 8.7 7.4 17.8
	(b 1 , b 2 , •) 42.2 9.5 4.7 14.0 17.1 19.2 9.6 6.7 22.6
	(b 1 , •, b 3 ) 41.8 9.6 4.6 13.8 17.1 19.0 9.7 6.6 22.4
	(•, b 2 , b 3 ) 42.0 9.5 4.4 14.0 17.0 19.2 9.8 6.8 22.4
	(b 1 , b 2 , b 3 ) 42.2 9.5 4.5 13.7 17.0 19.2 9.5 6.7 22.6
	no constr. 81.2 8.4 8.2 8.7 79.6 10.4 7.5 8.1 80.8

http://www.ncbi.nlm.nih.gov/genbank/

The query script is available upon request.

sequences in frame and for X0, in 4.6% of the 3248 sequences the coverage of the X0 code on the sequences read in frame is significantly smaller than those of the permutations. The results of Tab. 3 are somehow complementary to those of Tab. 2. In fact, from the no constr. row we see that the percentages of rejection for non natural combinations are high. This means that coding sequences in frame are made as to minimize the proportion of codons that belong to the codes X1 and X2, with a higher percentage for X2. This result is correlated with the unexpected code asymmetry between X1 and X2 observed in reading frames (see Fig. 2 in Arquès et al., 1997). Also, frame shift +1 and +2 sequences seem to minimize the proportion of codons that belong to the code X0 and X1

coding and non-coding regions in DNA sequences and its evaluation. J.

Theor. Biol.