From Time to Space Recurrences in Biopolymers

Abstract

The application of Recurrence-Based Techniques to biopolymers is herewith introduced with an emphasis on the differences holding between the analysis of strings endowed with a mainly logical (DNA) or chemico-physical (Proteins) information content. The unique features of RQA when applied to systems in which spatial order (sequence) takes the place of time are described, highlighting the emergence of ‘time distortions’. This is a metaphorical term stressing the fact that a monodimensional array of aminoacid residues (sequence) beside being formally identical to a discrete time series is a physical object that folds in the usual three dimensional space. This behavior allows to fully appreciate the fact that RQA as an analytical tool is flexible enough to deal with complex networks in either the spatial or the temporal dimension. The comparison of DNA sequences with text strings helps to shed light on the particular nature of biological information coding as well as on the role of RQA technique in bioinformatics and computational biology fields.

Appendices

Appendix 1: Cryptography

Cryptography was a strategically crucial discipline during the Second World War: the decipherment of hidden information in encrypted messages (like in the case of the cracking by the allies of the German code generated by the system Enigma) was based upon the notion that any human language, despite its apparent randomness and arbitrariness, is endowed with regularities of various kinds (e.g. the relative abundance of words of given length, the juxtaposition of pairs of symbols, etc.) and that no masking code can obscure the code-independent features typical of the original language. These code-independent features are supposed to derive from some general invariants common to all languages, like the so called Zipfs law [29] stating the frequency of occurrence of the words in any kind of (sufficiently long) text written in any language is negatively correlated with the number of letters according to a power law (Fig. 5.16).

Fig. 5.16

Scaling of the frequency of occurrence of words on word length as computed from different book collection

Figure 5.16 reports (on a double-logarithm scale) the strictly invariant relation between frequency of occurrence and word length in different book collections. Such regularities are clearly independent from the rich semantic information present in the analyzed books: the observed scaling comes from global constraints linked to the general features of human languages. The fact these general features are largely content independent was considered very important for many investigators involved in the analysis of the DNA sequences: in this way they could skip the specific (and extremely heterogeneous) function of different patches so to concentrate on the global statistical features of the billions letters DNA text.

Appendix 2: Strings from Human Languages

2.1 (A) Dante Alighieri - Inferno - I Canto (tercets 1–3) - FILTERED

Nelmezzodelcammindinostravitamiritrovaiperunaselvaoscuracheladirittaviaerasmarrita Ahiquantoadirqualeraecosaduraestaselvaselvaggiaeaspraefortechenelpensierrinovalapaura Tanteamarachepocoepiumortemapertrattardelbenchivitrovaidirodellaltrecosechivhoscorte

2.2 (B) Dante Alighieri - Inferno - I Canto (tercets 1–3) - NONFILTERED (English Translation by Henry Wadsworth Longfellow))

Midway upon the journey of our life

I found myself within a forest dark

For the straightforward pathway had been lost.

Ah me! how hard a thing it is to say

What was this forest savage, rough, and stern

Which in the very thought renews the fear.

So bitter is it, death is little more;

But of the good to treat, which there I found,

Speak will I of the other things I saw there.

2.3 (C) Dr. Suess Poem - NONFILTERED

I do not like eggs in the file.

I do not like them in any style.

I will not take them fried or boiled.

I will not take them poached or broiled.

I will not take them soft or scrambled,

Despite an argument well-rambled.

No fan I am of the egg at hand.

Destroy that egg! Today! Today!

Today I say!

Without delay!

(A), (B) and (C) refer to the strings from spoken languages whose % Det is shown in Fig. 5.14. Notice that in all cases RQA was applied after filtering the original texts as indicated in [30]: only in (A), however, the filtered text is shown. (A) and (B) show the first three of the 45 analyzed tercets in Fig. 5.14.

Appendix 3: Nucleotidic Strings

3.1 Satellite DNA 1 - GenBank: BI067039.1 Homo Sapiens Genomic Region Containing Hypervariable Minisatellites, mRNA Sequence

GTCCTCCGCCCCACACTTATGGGGCAGAACCCACACTTCCGGTCCTCCGCTCCACACTTATGGGGCACAGCCCACACTTCTGGTCCTCTGCCCCACACTTATGGGGCACAGTTGGGTGTTCTGCCCCACACTTATGGGGCACAGACAGCAGTTCCGGACCTCCACCCCACACTTATGGGGCAGAACCCACACTTCCGGTCCTCCGCCCCACACTTATGGGGCAGAACCCACAGTTTTGGTCCTCCGCTCCACACTTATGGGGCACAACAACCCACAGTTATGGGGCTTATGAGGTTCTGCCCCACACTTATGGGGCACAGACAGCAGTTCTGGTCCTCCGCCCCACACTTATGGGGCAGAACCCACACTTCCGGTCCTCCGCCCCACACTTATGGGGCAGAACCCACACTTCCGGTCCTCCGCTCCACACTTATGGGGCACAGCCCACACTTCTGGTCCTCTGCCCCACACTTATGGGGCACAGCTGGGGGTCCTACCCCACACTTATGGGGCAGAACCCACAGTTCCGGTCCTCCACCCCACACTTATGGGGCACAGCTGGGGATTCTGTGCCACACTTATGGGGCAGAACCCACAGTTCCGGCCCTCCGCCCCACACTTATGGGGCAGNNCNNGCNGNNCGGG

3.2 Satellite DNA 2 - GenBank: BM439581.1 Homo Sapiens Genomic Region Containing Hypervariable Minisatellites, mRNA Sequence

GGCACAGCTGGGGATTCTGCCCCACACTTATGCGGCACAACCCACAGTTCTGGTCCTCTCCCCCACACTTATGGGGCACAACAACCCACAGTTATGGGGCTTATGAGGTTCTGCCCCACACTTACGGGGCACAGACAGCAGTTCCAGTCCTCCGCCCCACACTTATGGGGCAGAACCCACAATTCCGGACCTCTGCCCCACACTTACGGGGCACAGCTGGGGATTCTGCCCCACACTTATGGGGCACAACCCACAGTTCTGGTCCTCTCCCCCACACTTATGGGGCAGAACCCACACTTCCGGTCCTCCGCCCCACACTTAGGGAGCAGAACCCACACTTCCGGTCCTCCGCCCCACACTTATGGGGCACAACAACCCACAGTTATGGGGCCTATGAGGTTCTGCCCCACACTTATGGGGCACAGACAGCAGTTCCGGACCTCTGCCCCACACTTATGGGGCACAGTTGGGGGTCCTACCCCACACTTATGGGGCAGAACCCACAGTTCCGGACCTCCGCCCCACACTTATGGGGCAGAACCCACACTTCCGNACCTCTGCCCCACACTTATGGGGCACA