Bacterial genomes lacking long-range correlations may not be modeled by low-order Markov chains: The role of mixing statistics and frame shift of neighboring genes

Quantifying local randomness in human DNA and RNA sequences using Erdös motifs

In 1932, Paul Erdös asked whether a random walk constructed from a binary sequence can achieve the lowest possible deviation (lowest discrepancy), for the sequence itself and for all its subsequences formed by homogeneous arithmetic progressions. Although avoiding low discrepancy is impossible for infinite sequences, as recently proven by Terence Tao, attempts were made to construct such sequences with finite lengths. We recognize that such constructed sequences (we call these "Erdös sequences") exhibit certain hallmarks of randomness at the local level: they show roughly equal frequencies of short subsequences, and at the same time exclude trivial periodic patterns. For the human DNA we examine the frequency of a set of Erdös motifs of length-10 using three nucleotides-to-binary mappings. The particular length-10 Erdös sequence is derived from the length-11 Mathias sequence and is identical with the first 10 digits of the Thue-Morse sequence, underscoring the fact that both are deficient in periodicities. Our calculations indicate that: (1) the purine(A and G)/pyridimine(C and T) based Erdös motifs are greatly underrepresented in the human genome, (2) the strong(G and C)/weak(A and T) based Erdös motifs are slightly overrepresented, (3) the densities of the two are negatively correlated, (4) the Erdös motifs based on all three mappings being combined are slightly underrepresented, and (5) the strong/weak based Erdös motifs are greatly overrepresented in the human messenger RNA sequences.

Genomic signatures in viral sequences by in-frame and out-frame mutual information

In order to understand the unique biology of viruses, we use the Mutual Information Function (MIF) to characterize 792 viral sequences comprising 458 viral whole genomes. A 3-base periodicity (3-bp) was observed only in DNA-viruses whereas RNA-viruses showed irregular patterns. The correlation of MIF values at frequencies of 3-bp (in-frame) with frequencies of 4 and 5bps (out-frame), turned out to be useful to distinguish viruses according to their respective taxonomic order, and whether they pertain to any of the three different kingdoms, Eubacteria, Archaea and Eukarya. The clustering of viruses was carried out by the use of a new statistics, namely, the pair of in- and out-frame values of the MIF. The clustering thus obtained turned out to be entirely consistent with the current viral taxonomy. As a result we were able to compare in a single plot both viral and cellular genomes unlike any given phylogenetic reconstruction.

Entropy and long-range memory in random symbolic additive Markov chains

The goal of this paper is to develop an estimate for the entropy of random symbolic sequences with elements belonging to a finite alphabet. As a plausible model, we use the high-order additive stationary ergodic Markov chain with long-range memory. Supposing that the correlations between random elements of the chain are weak, we express the conditional entropy of the sequence by means of the symbolic pair correlation function. We also examine an algorithm for estimating the conditional entropy of finite symbolic sequences. We show that the entropy contains two contributions, i.e., the correlation and the fluctuation. The obtained analytical results are used for numerical evaluation of the entropy of written English texts and DNA nucleotide sequences. The developed theory opens the way for constructing a more consistent and sophisticated approach to describe the systems with strong short-range and weak long-range memory.

Mappability and read length

Power-law distributions are the main functional form for the distribution of repeat size and repeat copy number in the human genome. When the genome is broken into fragments for sequencing, the limited size of fragments and reads may prevent an unique alignment of repeat sequences to the reference sequence. Repeats in the human genome can be as long as 10⁴ bases, or 10⁵ − 10⁶ bases when allowing for mismatches between repeat units. Sequence reads from these regions are therefore unmappable when the read length is in the range of 10³ bases. With a read length of 1000 bases, slightly more than 1% of the assembled genome, and slightly less than 1% of the 1 kb reads, are unmappable, excluding the unassembled portion of the human genome (8% in GRCh37/hg19). The slow decay (long tail) of the power-law function implies a diminishing return in converting unmappable regions/reads to become mappable with the increase of the read length, with the understanding that increasing read length will always move toward the direction of 100% mappability.

Entropy and long-range correlations in DNA sequences

We analyze the structure of DNA molecules of different organisms by using the additive Markov chain approach. Transforming nucleotide sequences into binary strings, we perform statistical analysis of the corresponding "texts". We develop the theory of N-step additive binary stationary ergodic Markov chains and analyze their differential entropy. Supposing that the correlations are weak we express the conditional probability function of the chain by means of the pair correlation function and represent the entropy as a functional of the pair correlator. Since the model uses two point correlators instead of probability of block occurring, it makes possible to calculate the entropy of subsequences at much longer distances than with the use of the standard methods. We utilize the obtained analytical result for numerical evaluation of the entropy of coarse-grained DNA texts. We believe that the entropy study can be used for biological classification of living species.