EXCEPTIONALITY OF A WORD AND ITS SUBWORDS

The exceptionality of a word under a given model may be related to the exceptionality of some of its subwords under the same model (see the comparison of exceptional words under two models); in this case, it results on a contamination phenomena. In other cases, it is an additional constraint for the sequence.

The pyramidal display is then very convenient to study simultaneously the exceptionality of a word and of its subwords.

A word W of length h can be represented by a pyramid of h-2 stages; each stage corresponds to a word length. The higher stage is composed with a unique square coloured according to the statistic of W. The stage beneath is made of two squares corresponding to the 2 subwords of length h-1 of W, and so on up to the stage made of h-2 squares associated with the subwords of length 3.

A pyramid can be made either under a single model (M1, for instance) or under maximal models for each stage (M1 for 3-words, M2 for 4-words, ...). Models with phase can be used.

For instance, let us look at the pyramids of some of the most under-represented 6-words (the first, the fifth and the sixth) in a sequence of E. coli (111,402 bases) under maximal models without phase.

• GGCGCC is the most under-represented 6-word in the sequence under M4: its square is black and its pyramid is topped by a minus sign. Its 2 subwords of length 5 are also under-represented under M3, whereas two of its 3 subwords of length 4 (GGCG and CGCC) are over-represented under M2. Subwords of length 3 are not particularly exceptional.

  When we analyze this pyramid from the bottom to the top we can say that:

  • GGC and GCG are slightly over-represented but they often occur side by side because GGCG is exceptionally frequent. The same for CGC and GCC.
  • GGCG is exceptionally frequent but GCGG is so rare that GGCGG is under-represented. In other words, GGCG is rarely followed by a G. Similarly, CGCC is over-represented but is rarely preceded by a G.
  • Finally, even when taking into account the rarity of GGCGC and GCGCC in the sequence, GGCGCC is under-represented. It means that the occurrences side by side of these two words are strongly avoided.

• When looking at the pyramid of AGCGCT, we can see that this under-represented 6-word is roughly composed of non-exceptional subwords. The exceptionality really starts with the 6 letters AGCGCT.

• TCCGGA is a very interesting word. Its 2 subwords of length 5 (TCCGG and CCGGA) are over-represented in the sequence but it seems that there is a stronger constraint: they "do not" have to occur side by side.

  

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