Abstract:
Kolmogorov-Arnold analysis is invented as a tool for the study of tumor and normal coding in human genomics sequencing to identify tumor-specific (somatic) sequences and copy number alterations. The technique enables to reveal quantitatively somatic sequences in the dataset of the genomic DNA purified from tumor and normal specimens. The computer power requirements for an analysis are modest ones.

Description:
SUMMARY OF THE INVENTION 
       [0001]    The Kolmogorov-Arnold technique is invented for the analysis of the statistical properties of the human genome and revealing the somatic sequences. This technique has been developed by Arnold during 2008-2009 based on the work of Kolmogorov of 1933, when he (Arnold) demonstrated its power on diverse examples of dynamical systems and number theory. The carried out applications for physical problems reveal the efficiency of the technique to separate signals with given correlations from composite signals, enabling to detect weak subsignals due to their correlation properties. The technique has been applied to astrophysical problems, i.e. for the analysis of observational datasets, e.g. on the cosmic microwave background radiation obtained by the Wilkinson Microwave Anisotropy Probe&#39;s (NASA), X-ray galaxy cluster data obtained by XMM-Newton satellite. The technique had enabled e.g. not only to reveal cosmic sources found by several other methods but also to predict others, then unknown, soon after discovered by the Fermi satellite. Recently this technique was instrumental for detection of the Yarkovsky-Rubincam (thermal thrust) effect for LAGEOS satellites during the study of Lense-Thirring effect predicted by General Relativity. The application to the human genome datasets enables to trace the somatic (tumor) sequences within the normal environment sequences. The application steps include: (a) modeling and extensive numerical experiments with sequences of the characteristics of the human genomic sequence: the revealing of the behavior of the stochastic parameter; (b) development of the criteria for the separation of particular signals, study of the sensitivity to the input parameters and the classes of those generated signals, estimation of their degree of randomness; (c) study of the systems with non-Gaussian cumulative functions; (d) the work with real genomic sequences. The analysis of a single genome sequence requires modest computer power. 
       OTHER PUBLICATIONS 
       [0000]    
       
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         Daruwala, R. S., Rudra, A., Ostrer, H., Lucito, R., Wigler, M., Mishra, B. (2004) A versatile statistical analysis algorithm to detect genome copy number variation. Proc. Natl. Acad. Sci. 101:16292-16297 
         Smyth G. K.: Limma: linear models for microarray data. In Bioinformatics and Computational Biology Solutions using R and Bioconductor. Edited by Gentleman R, Carey V, Dudoit S, Irizarry R, Huber W. New York: Springer; 2005:397-420; 
         Rakyan V K, Down T A, Thorne N P, Flicek P, Kulesha E, Gräf S, Tomazou E M, Bäckdahl L, Johnson N, Herberth M, Howe K L, Jackson D K, Miretti M M, Fiegler H, Marioni J C, Birney E, Hubbard T J P, Carter N P, Tavaré S, Beck S: An integrated resource for genome-wide identification and analysis of human tissue-specific differentially methylated regions (tDMRs),  Genome Res  2008, 18:1518-1529; 
         Ruike Y, Imanaka Y, Sato F, Shimizu K, Tsujimoto G: Genome-wide analysis of aberrant methylation in human breast cancer cells using methyl-DNA immuno precipitation combined with high-throughput sequencing, BMC Genomics 2010, 11:137; 
         Jeong Y-S, Kim D, Lee Y S, Kim H-J, Han J-Y, et al. Integrated Expression Profiling and Genome-Wide Analysis of ChREBP Targets Reveals the Dual Role for ChREBP in Glucose-Regulated Gene Expression. PLoS ONE 6(7) (2011); 
         Langmead B, Trapnell C, Pop M, Salzberg S L (2009) Ultrafast and memory-efficient alignment of short DNA sequences to the human genome. Genome Biol 10: R25; vol. 110 no. 19 J. A., et al′ Imprinted expression of genes and small RNA is associated with localized hypomethylation of the maternal genome in rice endosperm PNAS May 7, 2013, 110 no. 19 7934-7939; 
         Ewens W. J., Grant G., Statistical Methods In Bioinformatics: Springer Verlag, 2005; 
         Deonier R. C., Waterman M. S., Tavare S., Computational Genome Analysis: An Introduction, Springer Verlag, 2005; 
         Balding D. J., Bishop M., Cannings C., Weight Array Method, Handbook of Statistical Genetics Wiley, 2008. 
         Huang S, Yeo A A, Li S D., Modification of Kolmogorov-Smirnov test for DNA content data analysis through distribution alignment, Assay Drug Dev. Technol. 2007, 5, 663. 
         Hodgson D. R., Wellings R, Orr M. C., McCormack R., Malone M., et al. Circulating tumor-derived predictive biomarkers in oncology, Drug Discov. Today 15: 98-101 (2010); 
         Hicks J. et al, Novel patterns of genome rearrangement and their association with survival in breast cancer,  Genome Res . 2006.16: 1465-1479; 
         Closas P., Coma E., and Méndez L., Sequential detection of influenza epidemics by the Kolmogorov-Smirnov test, BMC Med. Inform. Decis. Mak. 12 (2012), 112; 
         Chen, Z., Liu, Q. &amp; Nadarajah, S. A new statistical approach to detecting differentially methylated loci for case control Illumina array methylation data. Bioinformatics 28, 1109-13 (2012); 
         Hansen, K. D. et al. Increased methylation variation in epigenetic domains across cancer types. Nat Genet. 43, 768-75 (2011). 
         Du X., Wojtowicz D., Bowers A., Levens D., BenhamT. C., Przytycka M., The genome-wide distribution of non-B DNA motifs is shaped by operon structure and suggests the transcriptional importance of non-B DNA structures in  Escherichia coli , Nucl. Acids Res. (2013); 
         Orlov Y. L., Potapov V. N., Complexity: an internet resource for analysis of DNA sequence complexity, Nucleic Acids Res. (2004) July 1; 32: W628-W633; 
         Ming. Li, P. M. B. Vitanyi, An introduction to Kolmogorov complexity and its applications, Springer-Verlag, 2008, p. 664; 
         Bin Li, Yi-Bing Li and Hong-Bo He, LZ Complexity Distance of DNA Sequences and Its Application in Phylogenetic Tree Reconstruction, Geno. Prot. Bioinfo, 3, 206 (2005). 
         U.S. Pat. No. 8,560,469 B2 (2013), Laine; Unto (Vantaa, FI), Rasanen; Okko (Kauniainen, FI), Method for a pattern discovery and recognition. 
         U.S. Pat. No. 7,809,765 B2 (2010), Evans; Scott Charles (Schenectady, N.Y.), Markham; Thomas Stephen (Schenectady, N.Y.), Torres; Andrew Soliz (Troy, N.Y.), Sequence identification and analysis. 
         U.S. Pat. No. 6,291,182 B1 (2001), Schork, et al., Methods, software and apparati for identifying genomic regions harboring a gene associated with a detectable trait. 
       
     
       BACKGROUND OF THE INVENTION 
       [0029]    A number of methods have been applied to study the statistical properties of genome sequences. Among the aims were the distinguishing of the coding regions, the exons, revealing of the signatures of particular features in the gene sequences, e.g. Bayesian model using Stochastic Search Variable Selection for genomic selection, statistics of consensus sequences for the most common nucleotides, Signals models, Weight Matrix Method, Markov models, multiple alignment of sequences (for given proteins), particular ways of splitting the sequences into two (phylogenetic partition). Kolmogorov-Smirnov method was among the applied ones. These and other methods inquire into various statistical features of the genome structure depending on the particular aim, which are far different from those addressed by KSP. Let us briefly outline the differences between the Kolmogorov-Smirnov test and the Kolmogorov (Kolmogorov-Arnold) stochasticity parameter (KSP) used below. The former is a long known test, while the second one has been developed by Arnold in 2008-2009. (Arnold does not even quote long known Kolmogorov-Smirnov method in his those papers.) Arnold defines KSP as an objectively measurable degree of randomness of observable events. That definition is similar that of Kolmogorov-Sinai entropy h(T) of Ergodic theory (Cornfeld, I., Fomin, S., and Sinai, Ya. G. Ergodic Theory. New York, Springer-Verlag, 1982): if h(T)&gt;0, then a dynamical system T is chaotic (simplifying the mixing/chaotic terminological link). However, and it is crucial, that h can be used to compare quantitatively different dynamical systems. Dynamical system T — 1 is said to be more chaotic than T — 2 if h(T — 1)&gt;h(T — 2)&gt;0. h(T)&gt;0 is the “Kolmogorov-Sinai test” (chaotic or not), h(T — 1)&gt;h(T — 2)&gt;0 is the “KSP test” (degree of randomness). Most importantly, KSP is applicable to even strongly correlated datasets (Arnold 2008, 2009). KSP technique has been applied to strongly correlated datasets of cosmic microwave background, considering those as not temperature 2D-maps but as 1D scalars, where the variable (temperature) is given by a random (Gaussian) field. The options are the consideration of (a) one sample from n-dimensional multivariate Gaussian distribution, or (b) n samples from (different) univariate Gaussian distributions. As for the Kolmogorov complexity (algorithmic information), it is also entirely different concept than KSP since deals with the minimal length of the coding string (computer program) for the Turing machine and hence is a unreachable quantity; see e.g. (V. G. Gurzadyan, 2005) for the complexity estimation of the human genome. Thus the Kolmogorov-Arnold technique provides novel possibilities to monitor the statistical properties of the genomic sequences, and to reveal somatic alterations. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0030]      FIG. 1 . The χ 2  for the Kolmogorov function for the sequence z n  vs α indicating the contribution of the random and regular parts for the pair of indicated input parameters. 
           [0031]      FIG. 2 . The dependence of characteristics of the χ 2 ′Δ, vs the parameter a. 
           [0032]      FIG. 3 . χ 2  frequency counts for the set of regular sequences as compared with a Gaussian function. 
           [0033]      FIG. 4 . The same as  FIG. 3  but for a set of random sequences. 
           [0034]      FIG. 5 . Kolmogorov function Φ for the sequences in  FIG. 3 . 
           [0035]      FIG. 6 . Kolmogorov function Φ for the sequences in  FIG. 4 . 
           [0036]      FIG. 7 . The maxima of χ 2  in  FIGS. 5 and 6  vs the number of the fluctuations. 
           [0037]      FIG. 8 . The function Φ for the rows with a sample of somatic (tumor) sequences (red) and normal ones (blue) averaged over the set of somatic ones and the 20 rows, prior and after the somatic ones. 
           [0038]      FIG. 9 . The same as in  FIG. 8  but for denoted somatic sequences (mutations). 
       
    
    
     DETAILED DESCRIPTION 
       [0039]    The invention is based on the Arnold&#39;s development of 2008-2009 (Arnold V. I., ICTP/2008/001, Trieste, 2008; Uspekhi Matemat. Nauk, 63 (2008) 5; Trans. Moscow Math. Soc. 70 (2009) 31; Funct. Anal. Other Math. 2 (2009) 139) of the work of Kolmogorov of 1933 (Kolmogorov A. N., G. Ist. Ital. Attuari, 4 (1933) 83). Consider X 1 , X 2 , . . . , X n  independent values of the same real-valued random variable X in growing order X 1 &lt;X 2 &lt;X n  and let (Kolmogorov 1933; Arnold 2008) 
         [0000]        F ( x )= P{X≦x}   
         [0000]    be a cumulative distribution function (CDF) of X. The empirical distribution function F n (x) is defined as 
         [0000]    
       
         
           
             
               
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         [0000]    Kolmogorov stochasticity parameter is 
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         [0000]    Kolmogorov theorem (Kolmogorov 1933) states that for any continuous F 
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         [0000]    the convergence is uniform and Kolmogorov distribution Φ is independent on CDF. The reasonable values of the function Φ correspond to the interval 0.3&lt;λ_n&lt;2.4; λ itself is a random variable with a mean 
         [0000]      λ mean =∫λφ(λ) dλ≈ 0.875029
 
         [0040]    This technique has been applied to the study of the properties of the datasets of cosmic microwave background radiation obtained by Wilkinson Microwave Anisotropy Probe (WMAP), including e.g. the study of the so-called Cold Spot and supporting its void nature, enabling the detection of point sources (quasars, blazars) in WMAP data later discovered by Fermi satellite, for detection of galaxy clusters based on X-ray data of XMM-Newton space telescope (V. G. Gurzadyan, A. A. Kocharyan, Kolmogorov stochasticity parameter measuring the randomness in the Cosmic Microwave Background; Astr. &amp; Astrophys. 492 (2008) L33; V. G. Gurzadyan, A. A. Starobinsky, T. Ghahramanian, A. L. Kashin, H. Khachatryan, H. Kuloghlian, D. Vetrugno, G. Yegorian, Large Scale Plane-Mirroring in the Cosmic Microwave Background WMAP5 Maps, Astr. &amp; Astrophys., 490 (2008) 929; V. G. Gurzadyan, A. E. Allahverdyan, T. Ghahramanyan, A. L. Kashin, H. G. Khachatryan, A. A. Kocharyan, H. Kuloghlian, S. Mirzoyan, E. Poghossian, Kolmogorov CMB Sky, Astr. &amp; Astrophys. 497 (2009) 343; V. G. Gurzadyan, A. L. Kashin, H. G. Khachatryan, A. A. Kocharyan, E. Poghosian, D. Vetrugno, G. Yegorian, The power spectrum of the cosmic microwave background Kolmogorov maps: possible clue to correlation of voids, Astr. &amp; Astrophys. 506 (2009) L37; V. G. Gurzadyan, A. L. Kashin, H. G. Khachatryan, A. A. Kocharyan, E. Poghosian, D. Vetrugno, G. Yegorian, Kolmogorov analysis detecting radio and Fermi gamma-ray sources in cosmic microwave background maps, Europhys. Lett. 91 (2010) 19001; V. G. Gurzadyan, A. E. Allahverdyan, T. Ghahramanyan, A. L. Kashin, H. G. Khachatryan, A. A. Kocharyan, S. Mirzoyan, E. Poghosian, D. Vetrugno, G. Yegorian A weakly random Universe?Astr. &amp; Astrophys. 525 (2011) L7; Gurzadyan, V. G.; Durret, F.; Ghahramanyan, T.; Kashin, A. L.; Khachatryan, H. G.; Poghosian, E., Detection of X-ray galaxy clusters based on the Kolmogorov method, Europhysics Lett., 95, 69001 (2011). V. G. Gurzadyan, I. Ciufolini, S. Sargsyan, G. Yegorian, S. Mirzoyan, A. Paolozzi, Satellite probing General Relativity and its extensions and Kolmogorov analysis, Europhysics Lett. 102 (2013) 60002.) 
         [0041]    Important class of sequences, particularly for the genomics problem, include those discussed by Arnold (quoted above papers, 2008, 2009) and represented in the form (V. G. Gurzadyan, T. Ghahramanyan, S. Sargsyan, Degree of randomness: numerical experiments for astrophysical signals, Europhys. Lett. 95 (2011) 19001) 
         [0000]        z   n   =αx   n +(1−α) y   n  
 
         [0000]    where x n  are random sequences and 
         [0000]    
       
         
           
             
               
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         [0000]    are regular sequences, a and b are mutually fixed prime numbers; both sequences are defined within the interval (0,1) and have uniform distribution, α indicating the fraction of random and regular sequences. When doing statistic with large number of sequences, each new sequence y n  is taken as the continuation of the previous one from the same arithmetical progression. 
         [0042]    Thus one has z n  with a distribution function 
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         [0043]    Analyzing the stochastic properties of z n  vs the parameter α varying between 0 and 1 for different values of the parameters a and b, i.e. corresponding to from purely stochastic to purely regular sequences, the  FIG. 1  can be obtained.  FIG. 1  shows the χ 2  for the empirical distribution functions G(Φ) m  and G 0 (Φ)=Φ as indicator for randomness. Thus one parameter χ 2  is calculated per each of the (100×101) sequences. Grouping 100 χ 2  values per one value of α, one can construct the mean and error values for χ 2 , resulting the behavior for each pair of a and b at the dependence of χ 2  on α. Thus the variation of χ 2  vs α varying from 0 to 1 indicates the gradual change of sequences z n  from regular to random. 
         [0044]    That sensitivity on the parameter a on b is defined via the parameter Δ which equals the difference of two values in the χ 2  plots: maximal value of χ 2  and minimal value in the range α from (0, α_{max}), where α_{max} is the position of the maximal value.  FIG. 2  shows the strict mirror symmetry in the dependence of Δ vs a, although no periodicity has been found by Fourier analysis. 
         [0045]    The features of signals formed as sum of many fluctuations, each having the same standard deviation, with the χ 2  frequency counts for the set of random and regular sequences as compared with a Gaussian function, also reflect the informativity of the method. 
         [0046]    Consider a regular sequence given as 
         [0000]    
       
         
           
             
               
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         [0047]    Where χ k  is compactified arithmetical sequence within the interval (−1,1), with step 1/k, and 
         [0000]    
       
         
           
             
               
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         [0000]    as a random one. Compact(x,p,q)=q+x mod(p−q) indicates multiples of (p−q) from x having the value within the range (p,q), p&lt;q. 
         [0048]    The results for 10000 random (generated by random number generator) and regular sequences each are shown in the two  FIGS. 3 and 4  for χ 2  with varying the number of the fluctuations. 
         [0049]    It is seen from the  FIGS. 5 and 6 , that although Gaussians are maintained both in random and regular cases, the behavior of Φ is radically different, thus revealing the possibilities of this method. 
         [0050]    One more indication shown in  FIG. 7  is the revealed weak dependence to the number of fluctuations. 
         [0051]    Genomics Application. 
         [0052]    The objective is the sequencing analysis of samples to identify tumor-specific (somatic) sequence and copy number alterations. The procedure is illustrated below via the dataset of the genomic DNA purified from tumor and normal specimens. DNA samples were enriched for coding regions in the genome using custom DNA capture approaches. Sequence data were mapped to the reference human genome sequence and sequence alterations were determined by comparison of up to 57 million 100 digit rows of G,A,T,C bases of tumor and normal DNA. The study of somatic sequence alterations has been performed via the Kolmogorov-Arnold analysis for the dataset. Input data, i.e. the sample rows, with indicated row&#39;s number, are shown below: 
         [0000]    
       
         
               
             
           
               
                 30259203 0 
               
               
                 TATGTCCCCCAATAAACTGTAAGCTTCAGGGGGAATGAGTGAATGACCAG 
               
               
                 TGTTCTAAGTAGAATCTTAATCTTGGATTGTTAACTCAATGAATTCTCAT 
               
               
                   
               
               
                 30259204 0 
               
               
                 ATGTGGTCCCTGGGCTTCTCCATTCACCAGCCCCCAATCATTTCTTCCTG 
               
               
                 TACAAGGAATGAAACAGTCATTGAGGTCACCTCTAGCAAGTCTCCCAGCA 
               
               
                   
               
               
                 30259205 0 
               
               
                 GATAGCTACCACTTTCCACATCACAGTAGTAATAAATTAAATCATAATATA 
               
               
                 CTTAATATTAGGTGTACTTTTTAAAACTAAATTTTAAAAGAAAAAGGAG 
               
             
          
         
       
     
         [0053]    The sample of 10 somatic sequence signatures have been studied for a set of data (each string of over 5000 mln nucleotides), and the presence of each somatic sequence can be summarized as follows. 
         [0000]    
       
         
               
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                   
               
               
                 Gene 
                 Sequence 
                 N 
                 81 
                 82 
                 83 
                 84 
                 121 
                 122 
                 150 
                 151 
               
             
          
           
               
                 Symbol 
                 Context 
                 all 
                 N1 
                 T1 
                 T2 
                 N2 
                 N1 
                 N2 
                 T1 
                 T2 
                 N2 
                 N2 
                 T2 
                 N2 
                 T2 
               
               
                   
               
               
                 ABP1 
                 GAGCCNAAGAC 
                 1 
                   
                   
                   
                   
                   
                   
                   
                 1 
                   
                   
                   
                   
                   
               
               
                 AOX1 
                 AGGCCNGCCCT 
                 3 
                 1 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 1 
                   
                 1 
               
               
                 APOB 
                 GATCTNAAAAA 
                 2 
                   
                   
                   
                 1 
                   
                   
                   
                   
                   
                   
                   
                   
                 1 
               
               
                 ATRX 
                 GTGAGNACGTT 
                 1 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                 1 
                   
               
               
                 CLASP2 
                 GTGACNCATAG 
                 1 
                   
                   
                   
                   
                 1 
                   
                   
                   
                   
                   
                   
                   
                   
               
               
                 DDX27 
                 AAAACNTCTCG 
                 0 
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
                   
               
               
                 DOCK9 
                 TGCCTNTGGAT 
                 4 
                   
                   
                   
                   
                   
                   
                 1 
                   
                 1 
                   
                   
                 2 
                   
               
               
                 hFAM131B 
                 TCATGNCTGTG 
                 3 
                   
                 1 
                   
                   
                   
                   
                   
                   
                   
                 1 
                   
                   
                 1 
               
               
                 FASTKD3 
                 TCAAGNTGATT 
                 3 
                   
                   
                 1 
                   
                   
                 1 
                   
                 1 
                   
                   
                   
                   
                   
               
               
                 FZD7 
                 GACTGNGGTAT 
                 0 
               
               
                   
               
             
          
         
       
     
         [0054]    An example of the presence of the somatic sequences in two samples of this Table is shown below. 
         [0000]    
       
         
               
             
           
               
                 084N2 
               
               
                 34588855 0 
               
               
                 ACAGAGCTGAGTCACGTGGTGGAATAAATACAAGTGTGTTGGCTCAAGCC 
               
               
                 CAGAGTTGCTGATAGGGAGATGGATGCCTCCTGCCCCATGGGCTGTAAAC 
               
               
                   
               
               
                 34588856 0 
               
               
                 TGCCTGGAACATGACCCTCCTAGACCAACTCCACACTGGACTTCATCAGC 
               
               
                 AACTGCAACACCTGGAGACCTGCTTGCTGCAGGTAGTGGGAGAAGGAGAA 
               
               
                   
               
               
                 34588857 0 
               
               
                 GCTCAGAGGAGTGGGAATGGGGTGGGTCTTCCTGGGCATGGCCAGGCTGT 
               
               
                 GCCCCAGCCTAGGGGCTTGGTGTGGTGGTGTTGGCCAGGCTTGGCAGGCA 
               
               
                   
               
               
                 34588858 0 
               
               
                 TAGTAACTCTGATATGCAGTTGGGTAATTTAAAAATTGTAGTTTAAGTGG 
               
               
                 TTTCTGCACGAATCTAGATTCTCTAAATGAAATCTAACTCAGCTCCCATA 
               
               
                   
               
               
                 34588859 0 
               
               
                 TTCCACTGCTTAAACTGAGCACAGGACGTGCACTTGGATAGTAAACCAGG 
               
               
                 TGTCTCCTCAAAGCCCTAATATATTCAGCATCTCTATCAAAGGCGCCTTT 
               
               
                   
               
               
                 34588860 0 
               
               
                 GAGAAAGACTTAAGCATGATGCCCCCCCGCACTGACTTTCATGTGGCAGC 
               
               
                 CAAGCAAGCGCCGGACTCCTCAAGGGCTGGCAGGAGAGCAACAGGAGATG 
               
               
                   
               
               
                 34588861 0 
               
               
                 TTCCAGCCTTGGGCATAGTCAGGGACTCTCGTGGGACCCCCCGAGTAGAA 
               
               
                 ACACAGATGTGTCTCCCTGGGTCCCTGCCAGGTCCCCTCTCAGCCTGGAT 
               
               
                   
               
               
                 34588863 0 
               
               
                 CCATTGAAACCAATGCTATGCATACCACTCCTCCTTCCAAGGCTGTCATC 
               
               
                 TTGCAGTCTGCCTCTGTCTCCAAGAAGTACCTCCCATTTTACCTGGGGCT 
               
               
                   
               
               
                 34588866 0 
               
               
                 CCCGAGCCTCTCCTCGTGGGGAAGGCGACACACAGCAGCCGTGAAGGTGG 
               
               
                 CTGCCGAGAAGTTGCCCAGGCTGCTTCCAAAACCCACGGCCTGCTCCACG 
               
               
                   
               
               
                 34588867 0 
               
               
                 GTTTTGAACATTTGGCAATGTGTATTAGCACAGCTCTTCCCTTTAGCAAG 
               
               
                 ATTTGTTTTGCGGTTCCAGATAGGTCATTTCCAGGCCTCATTCCGCCAAG 
               
               
                   
               
               
                 345888687 
               
               
                 NNNNNGGTTAACCAAGGAAAAAAACTTAAAAGGCATCATTGTAAACA 
               
               
                            TGCAGAAGTTGAATNATTGATTTGAAAGTTTTAAT 
               
               
                 TTTGTAA 
               
               
                   
               
               
                 34588869 0 
               
               
                 TTCTCTATTTTCCTAGTTATATATGCTATCATATGTCTGTTTTTCTCCTC 
               
               
                 TTGAAGTTTCCCTGAAACCTGGGCTCTTGAAGACGCATCACTGGAGCAGA 
               
               
                   
               
               
                 34588870 0 
               
               
                 CCTCAGAGATGTTGACCCCAGCTGGGACATCAGTACCACACGCCACAAAA 
               
               
                 TCCCCATGTCAATAGCACCACCCCTTCCCTCTGCTGGAGGATACAATCCC 
               
               
                   
               
               
                 34588872 0 
               
               
                 AGCCCCACAGAGGGATCCCTTACTACTGGAATGCAGACACAGACGTTGGA 
               
               
                 TCCTGGTTCTCCCCAGATGACACCAACTCCGTGGTTACCAAATAGGCAAA 
               
               
                   
               
               
                 34588873 0 
               
               
                 CCCTTCCTCTGTCAGTACTGCTGTGAACAGACCTGCGTGGCTGGTCCTGT 
               
               
                 GTTTGGGTAAAACATTAAATGACAACTTTAATTGAAAAAACCTGGGGAAG 
               
               
                   
               
               
                 34588874 0 
               
               
                 GCCAGAACACTGATTGGTGGATACCCTCGTCTTCAGTCTCTCTGGAGTCC 
               
               
                 TTTCACTTCCATTGCCTTCGGGACAGGGAACACCTCCTCCCCCACCCTCA 
               
               
                   
               
               
                 34588875 0 
               
               
                 CCCCAGGGAGGGCAACGTTCATTCTGCTGTGGCAATGGGCCTTGGGGGAT 
               
               
                 AGGAGTTGGATATCTGCGTAAGGGAGGACAGGTGGAAAGTGATATGGAAC 
               
               
                   
               
               
                 34588876 0 
               
               
                 GTGCGTAAAATGACAAGTCAAGGAACGACAGGAAGCAGGGAAAGTCGACA 
               
               
                 CACTGCAAGAAAAATAGTGATAATGCTAAATAAAAAACCGTCATGAGAAA 
               
               
                   
               
               
                 34588887 0 
               
               
                 CATGCACCCCAACTAAGCTCAGGTCCTGCGTAGATTTTTGTGGGAATCAG 
               
               
                 CTTCCAAGGACTAGGTCATTTATTGGCCCTGTGTTCCTAGGTCAGCGGTG 
               
               
                   
               
               
                 34588888 0 
               
               
                 GAGGTATGGTCAACCTATAGTAAGTTAGTAAATGATAAGGGGAGGAAGAA 
               
               
                 ATGGAAACCTAAACATCTACTGCAATGAAAACCAACAGCAATATCAGGAG 
               
               
                   
               
               
                 34588889 0 
               
               
                 AGAACAAACAAAACTATAGTGCTGGACACAAGTGGAAAGTATATTACCTT 
               
               
                 CATTAAAATTATCTTCCTGACATGCAGTGAGGAAATTGTGTTTCCAAAGA 
               
               
                   
               
               
                 151N2 
               
               
                 30259182 0 
               
               
                 CCCAAGCTGGAATGCAGTGGCTGTTCACAGATGCAGTCATAGCACTGTAG 
               
               
                 AGCATCAGGTGTTCCTCCCATGGCAGGAGGGAAGAAACCCAGCAAACGTC 
               
               
                   
               
               
                 30259183 0 
               
               
                 AATGAAAGTCAAATGAAATCTTTAGATGTCCTTCCATATTTTATGGTGAC 
               
               
                 GATCTGCATCATGGGTCCGGCGGGTCCGGCCCAGGCAGCCTCTGGAGCAA 
               
               
                   
               
               
                 30259184 0 
               
               
                 AAATGGGATACAGAAAAAGACAGAACATCATAGTTCTAAGTTACTGATTT 
               
               
                 GGAGTTTCACTCTTGTTGCTGAGACTGGAGTGCAATGACGCGATCTCGGC 
               
               
                   
               
               
                 30259185 0 
               
               
                 GACTCACTGGAGGAATGTGCCATCACTTGTTCAAATAGCCACGGCCCTTG 
               
               
                 AGTGAAGAGGCTGGGGATGACACTACCACTGTACTGTTCTGGCATGCTCT 
               
               
                   
               
               
                 30259186 0 
               
               
                 TTTTTTCCTTTCTTTCCTTTAGAAGAAGTAGATGAACGAGACGATGCAGC 
               
               
                 TCTGCCATTCAAGGCTCTAATGTTCCTAACCCTGATGAGGCCGTCCCTAG 
               
               
                   
               
               
                 30259187 1 
               
               
                 TGGCATCACTGAGGCACTGTACGGTCCTTGCAATGTCAAACTTACTCAAT 
               
               
                 AGTTNTCTGATTGCTTCCCCTTCCATAGGTTGTGGGAATAATTGAGTCAC 
               
               
                   
               
               
                 30259188 0 
               
               
                 CCAAGAAGGCCTTGACCTTTCCAGCAAGTGGGAAGGTATAATCAGTCTCC 
               
               
                 TAGAGCAGGACTCACCTGTTTTAACTCCACCTCCCGCCCCAACCCCAGTG 
               
               
                   
               
               
                 30259189 0 
               
               
                 GAGATCATTCGCATAAAGGGCCTGGGCCACTGAACGGGCTAAGCAAATAT 
               
               
                 GAATTATTTGGATCGGTTGGCAGAAGAGGTCAATGATAAATTGCAAGAAA 
               
               
                   
               
               
                 30259191 0 
               
               
                 ACTCTACTTTGAGGACAGTCCGGCTCTCAAGGAGCACGTCTTCCACAACC 
               
               
                 TGATCTCTCTTTCTTTACCCTACATCCAAGGGCTTGAACATTTCTTGACT 
               
               
                   
               
               
                 30259192 0 
               
               
                 GCCTTTGGGAGACCCTCTTCAGATGACTGCCATCAAAAGAGTCATAGAAA 
               
               
                 GCTGTGCAGGGCATCCCAGACAGGGGTTGGGCCTCCCTGATGGACAAGGA 
               
               
                   
               
               
                 302591945 
               
               
                 CGCCTCCCTATTCCTCTNNNANCAGGTCTCAAG          GCCT 
               
               
                 TCCCTCACCCCCTCACAGTGGTGCAGTTCTCGGCCTCTCGGCCTGGCTTC 
               
               
                   
               
               
                 30259193 0 
               
               
                 TCTATGTATGTCACATGCACTTGCTTAACTCTAGGCATGAAGAACTTCTC 
               
               
                 GAATGAATACAGAGTTCCTGAACTGAATGTTCAAAATGGAGTGTTAAAAT 
               
               
                   
               
               
                 30259196 0 
               
               
                 ACTTCTAATTTTTTTCTAACAGGCTCCACAATCTCTCATATATGTTTTTT 
               
               
                 TGTTATGTTGACACTGAAGTGGAAATGAAAGAAATCTATACAAGTAACCA 
               
               
                   
               
               
                 30259197 0 
               
               
                 GAGGAAGAACCTGCAGGAGCCATCCCTGACTTGCTTTGACCAAGCCCAAG 
               
               
                 GTTTAAAAGTTTGTCCCAGACCTCCAGTTATAATGCTTCTATGGCAGATT 
               
               
                   
               
               
                 30259198 0 
               
               
                 TAGATTCTGCTGCAGTTGTTCTTCTTTTTGTATAAATAATCTTCTTTGTT 
               
               
                 TGGGCACACCAGCTGTTGCCTTCTGGTGGGTGGGGCTGGGTGGAGCCACC 
               
               
                   
               
               
                 30259199 0 
               
               
                 TTTAGACGTTCTACATCAATAGCTCCCAATTTGAGGTACTTGGGCTGCTT 
               
               
                 TCCTGACAGCTTGGGGTTTGAGACCCTGGAAAACTGCAGGAAGGTACCGT 
               
               
                   
               
               
                 30259200 0 
               
               
                 GATTCAGGTGCATTTGTTTCAAGCCTTGAGGTTTCATGGGGCAAATGATG 
               
               
                 TTGAAGTTTAAGAATATCAACGTTTTATATGGTATTGTGTTTTTAGGAAC 
               
               
                   
               
               
                 30259201 0 
               
               
                 TCTCCTGTATGAGTTGTCTGATGTTGAGTAAGGCATGAACTACGGTTAAA 
               
               
                 ATGCCCAACACCTGGCTTGCCCACCCTCCTGTCCCCTGCCCCTTCCTGGT 
               
               
                   
               
               
                 30259202 0 
               
               
                 ACACACAATTGTATAGATTGCCAGCTTTTTTATAAACTGGAAATTTCTTG 
               
               
                 AAATCATCCTACCAAAAAGACATACATGCTCGCATGTTCACCACAATACT 
               
               
                   
               
               
                 30259203 0 
               
               
                 TATGTCCCCCAATAAACTGTAAGCTTCAGGGGGAATGAGTGAATGACCAG 
               
               
                 TGTTCTAAGTAGAATCTTAATCTTGGATTGTTAACTCAATGAATTCTCAT 
               
               
                   
               
               
                 30259204 0 
               
               
                 ATGTGGTCCCTGGGCTTCTCCATTCACCAGCCCCCAATCATTTCTTCCTG 
               
               
                 TACAAGGAATGAAACAGTCATTGAGGTCACCTCTAGCAAGTCTCCCAGCA 
               
               
                   
               
               
                 30259205 0 
               
               
                 GATAGCTACCACTTTCCACATCACAGTAGTAATAAATTAAATCATAATAT 
               
               
                 ACTTAATATTAGGTGTACTTTTTAAAACTAAATTTTAAAAGAAAAAGGAG 
               
               
                 
                           
                 
               
             
          
         
       
     
         [0055]    Kolmogorov function Φ(λ) has been obtained based on the codon coding for the row (string) containing the somatic sequence (mutation) and the average for 10, 20, 50 rows without mutations distributed prior or after the rows with mutations, with varying number of digits in the rows. The histogram in  FIG. 8  below shows the results for the subsequences of 100, 50 and 25 digits, i.e. nucleotides (noted as ‘word’). 
         [0056]    The function Φ reveals the difference in the coding for somatic/(tumor(indicated as with MC, black) and normal (gray) genome sequences. The averaging is for 119 files in the particular blocks of dataset with a sample of somatic sequences. The error bars, as expected, decrease for smaller sequences.  FIG. 9  exhibits the Kolmogorov function for a sample of somatic sequences. The CPU time for one sequence (about 10 9  nucleotides) was about 1 hour for i7, 2600 3.4 GHz processor of 6 GB memory. 
         [0057]    The results of the modeling of generated systems on the one hand, and the stability of the results of the real genomic strings with respect the number of the averaged rows, namely, for row numbers e.g. 10, 20, 50, 70, and their mutual sequences on the other hand, do confirm the efficiency of the method for such strings. 
         [0058]    This demonstrates that Kolmogorov&#39;s analysis is sensitive to genome coding differences at least for a sample of somatic/tumor specific vs normal sequences, and thus can act as a tool for detection and study of corresponding genome coding properties. 
         [0059]    Our approach differs from those using the Kolmogorov-Smirnov test, such as in U.S. Pat. No. 8,560,469 B2 (2013), U.S. Pat. No. 7,809,765 B2 (2010), U.S. Pat. No. 6,291,182 B1 (2001), since we deal with the stochasticity parameter as degree of randomness as defined by Arnold (2008,2009), the theoretical and empirical codon coded distributions, and obtain semi-robust quantitative characteristics for the mutation and normal (without mutation) genomic sequences.