Factorizable Language: Examples from Biology

1. Factorizable Language:Examples from Biology Bailin Hao Institute of Theoretical Physics, Academia Sinica, Beijing T-Life Research Center, Fudan University, Shanghai The Santa Fe Institute, New Mexico http://tlife.fudan.edu.cn/ http://www.itp.ac.cn/~hao/ CSSS2007 Beijing

2. NowCoarse-grained biological data → DNA (RNA) sequences→ Protein sequences

3. The Central Dogma of Molecular Biology replication DNA DNA reverse transcriptiontranscription cDNA mRNA translation Protein/Enzyme folding Function Structure interaction

4. Biological Symbolic Sequences • DNA : 1D, directed, unbranching heteropolymers made of 4 kinds of bases (a, c, g, t). Typical length: 104 to 108 nt • RNA: 1D, directed, unbranching heteropolymers made of 4 kinds of bases (a, c, g, u). Typical length: 20 to 104 nt • Proteins: 1D, directed, unbranching heteropolymers made of 20 kinds of amino acids. Typical length: 102 to 104 AA

5. Example of DNA Sequences From a tiny part of the rice genome (Beijing Genomic Institute, CAS) Draft sequences: 2002 Fine maps: 2005

7. Chloroplast Genome of Rice Originated from photosynthetic bacteria • Made of a、c、g、t (nucleotides) • Japonica (1989)：134525 nt; Indica (2001)：134559 nt • Very close to each other

8. 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9. Most of the Genomic DNAis Transcribed! • Protein-coding genes • Non-coding genes (various RNAs) • Many many controlling signals, e.g., TFBSs • Transposable elements: the real selfish genes • Various repeated segments, short and long • Not much “junk” DNA

10. From Eric Davidson, Science 1998

11. Sequencing Projects Worldwide(15 July 2007) • Published complete genomes: 614 (540 prokaryotic） • On-going prokaryotic projects：60 Archaea + 1270 Bacteria => 1330 • On-going eukaryotic projects: 773 • On-going metagenome projects：95 • Total: 2813 • http://www.genomesonline.org (GOLD DB)

12. GenBank Rel. 160(15 June 2007) • Sequences: 73 078 143 • Nucleotides: 77 248 690 945 bases • Average length: 1067

13. UniProt Rel. 11.2 (26 June 2007) • 4 736 514 protein sequences including 272 212 in SWISS-PROT (Rel.53.2) and 4 464 302 in TrEMBL • UniParc: 15 038 304 proteins In SWISS-PROT: • Min: 2 AA • Max：34 350 AA AA：Amino Acids

14. Example of a small protein： Light chain of human immunoglobulin ID A1BG_HUMAN STANDARD; PRT; 495 AA. ... ... ... KW Immunoglobulin domain; Glycoprotein; Plasma; Repeat; Signal. ... ... ...SQ SEQUENCE 495 AA; 54209 MW; 87A50C21CE89459C CRC64; MSMLVVFLLL WGVTWGPVTE AAIFYETQPS LWAESESLLK PLANVTLTCQ ARLETPDFQL FKNGVAQEPV HLDSPAIKHQ FLLTGDTQGR YRCRSGLSTG WTQLGKLLEL TGPKSLPAPW LSMAPVPWIT PGLKTTAVCR GVLRGETFLL RREGDHEFLE VPEAQEDVEA TFPVHQPGNY SCSYRTDGEG ALSEPSATVT IEELAAPPPP VLMHHGESSQ VLHPGNKVTL TCVAPLSGVD FQLRRGEKEL LVPRSSTSPD RIFFHLNAVA LGDGGHYTCR YRLHDNQNGW SGDSAPVELI LSDETLPAPE FSPEPESGRA LRLRCLAPLE GARFALVRED RGGRRVHRFQ SPAGTEALFE LHNISVADSA NYSCVYVDLK PPFGGSAPSE RLELHVDGPP PRPQLRATWS GAALAGRDAV LRCEGPIPDV TFELLREGET KAVKTIPTPG AAANLELIFV GPQHAGNYRC RYRSWVPHTF ESELSDPVELLVAES //

15. A Proviso: Limitations of Symbols • DNA sequences: no methylation info (Human Epigenome Project), no role of metallic ions, no spatial configurations, etc. • Protein sequences: no post-translational modifications (glycosylation, phosphorylation, splicing), no explicit structural info, etc.

16. These sequences are results of billions years of evolution • They contain conserved but variable, fuzzy deterministic/regular as well as random elements • The relation between random and non-random aspects may be scrutinized in different ways, e.g., in a collection of many sequences or in a single long sequence

17. A simple-minded approach:Counting K-strings • Not to define complexity measures (not appropriate for short strings) • K-words, K-grams, K-tuples, K-strings. (“string” versus “word”) Let us look at K-tuples in a collection of sequences or in a single long sequence

18. Transition from Randomness to“Determinism” with K Increasing • E. coli (strain K12) genome: a loop of 4639221 bp, distribution of single letters almost random • K=1, 2, …close to random sequences • Among the 44639221 possible sequences only 1 or a small subset makes E. coli (K12) • Probes on gene-chips or primers for PCR: K in the tenths, say, 20 ~ 80. Already gene-specific. • Some extreme examples follow

19. Genus-Specific Oligo-Nucleotides • The 18-tuple gttccaataagactaaaa appears only in the genomes of 3 species of the Archaea genus Pyrococcus: P. horikoshii, P. abyssi, and P. furiosus (as repeats in Non-CDS) • No other exact match in GenBank as of 17 July 2007 • It is a genus-marker if restricted to bacteria genomes only

20. A 25-string specific for 2 Archaea • aaatcagaccaaaatgggattgaaa • 107 copies in Archeoglobus fulgidus (A2.7) 171 copies in Methanobacterium thermoautotrophicus (A2.1) • The two species belong to different classes of A2 • Function unknown and annotation questionable

21. A 25-string specific for 2 Archaea • aaatcagaccaaaatgggattgaaa • The 25-string has no exact match in GenBank except from these two species themselves (as of 17 July 2007) • These 25-strings appear as clusters of almost periodic repeats

23. Species-Specific Oligo-Peptides • HAMSCAPDKE: only in E. coli and Shigella among more than 1.3 mil. known proteins in PIR database • HAMSCAPERD: only in Samonella • It is known: D (Aspartate) and E (Glutamate) are interchangeable in many homologous proteins K (Lysine) and R (Arginine) are interchangeable in many homologous proteins • A consensus HAMSCAP(D/E)(K/R)(D/E) picks up only Enterobacteria (a bacterial family in the Proteobacteria phylum, major players in human digestive tract)

24. Counting, Statistics and Combinatoricsof K-tuples (with not-too-big K) • New findings and surprises • Interesting mathematical problems (and solutions to some problems) • Meaningful biological knowledge obtained • A few examples from our own work

26. 2D Histogram of K-Tuples • Old version seeDNA implemented at NIST: http://math.nist.gov/~Fhunt/GenPatterns/ • Our updated software SeeDNA has been made public: http://www.itp.ac.cn/~hao/SeeDNA.tar.gz Genomics, Proteomics and Bioinformatics, 2(2004)1-19 • Relation to “Chaos Game Representation” of DNA: P. Tino, “Multifractal properties of Hao’s geometric representation of DNA sequences”, PhysicaA304 (2002) 480-499.

27. g c a t

28. Location of ctag-tagged strings

29. Avoided Strings in BacterialComplete Genomes Species-specificity of “avoidance signature”

30. Comparison with the Bergey’s Manual

31. Problem of True and RedundantAvoided K-Tuples in DNAs • DNA segments containing the substring ctag are under-represented in E. coli • At K=7 one missing (gcctagg) • At K=8 a total of 173 string types are missing • However, 173 = 8 + 165 8 redundant and 165 true at K=8 • How to get the exact solution in general?

32. True and redundant Avoided Strings • Suppose at length K one K-string S was missing • Strings xS and Sy must be missing at length K+1.There are 8 of them. • Strings xyS, xSy, Sxy cannot appear at length K+2. There are 16*3=48 such strings. • By induction: at length K+i the number of missing strings is 4i (i+1). • Not an exact answer: self-overlap ignored.

33. Two Mathematical Problemsin 2D Histograms • Dimensions of the complementary sets of portraits of tagged strings. • Number of true and redundant missing strings. The two problems turn out to be one and the same, the first being graphic representation of the second.

34. Two Methods to Solve the Problem • Combinatorial solution: Goulden-Jackson cluster method (1979); number of dirty and clean words. • Language theory solution: factorizable language, minimal deterministic finite-state automaton.

35. Σ* = L + L’ L = Σ* - L’ L’ = Σ* - L Admissible and Inadmissible words Language L and its complement L’

36. Factorizable Language • A language L is called factorizable if any substring of a word in L also belongs to L • Factorizability leads to the existence of a set of (minimal) forbidden words • A factorizable language is determined by the set of (minimal) forbidden words • A factorizable language may be recognized by Finite State Automata (FSA)

37. Σ* = L + L’ L’ = Σ*L’’ Σ* L = Σ* - L’ = Σ* - Σ*L’’ Σ* A factorizable language L is defined by L’’ Factorizable language L: a core L’’of (minimal) forbidden words in L’

38. Define a Factorizable Language L(G)By a complete genome G • Take many identical copies of the genome G • Cut them in all possible ways: from single nucleotides, dinucleotides, trinucleotides, …, to the whole uncut genome • Collect all these strings and add an empty string → Language L(G) • L(G) is factorizable by construction

39. 1D Histogram of K-Tuples • Collect those K-tuples whose count fall in a bin from to ; • Plot the number of such K-tuples versus the counts. • This is a 1D histogram or an expectation curve if one wants to calculate it from a statistical model

40. 1-D Histograms (continued) • X-axis: string counts from 0 to some maximal number (with K fixed) • Y-axis: number of different string types within a count bin: 0 (absent or avoided string), 1-3, 4-6 (rare strings), … to some big numbers, e.g. 774 (repeats)

42. Effect of Randomization • What happens to the 1D and 2D histograms if we randomize the original genomic sequence? • One must do this comparison in order to show that the somewhat regular patterns seen in the bacterial portraits are not incidental.

44. A Surprise in 1D Histograms of K-Tuples of Randomized Prokaryote Genomic Sequences A single biased peak with a long tail in the original genome Rich fine structure in randomized sequences of some genomes

49. G+C Content of Some Bacteria

50. Fine Structure Caused by BiasedG+C Content • There are K+1 peaks in a 1D histogram of K-strings from a randomized genome • Each peak is described by a Poisson distribution whose parameter is determined from the G+C content and the sequence length • Even the single peak of randomized E. coli genome is a superposition of K+1 Poisson distributions