Hierarchical multifractal representation of symbolic sequences and application to human chromosomes.

The two-dimensional density correlation matrix is constructed for symbolic sequences using contiguous segments of arbitrary size. The multifractal spectrum obtained from this matrix motif is shown to characterize the correlations in the symbolic sequences. This method is applied to entire human chromosomes, shuffled human chromosomes, reconstructed human genomic sequences and to artificial random sequences. It is shown that all human chromosomes have common characteristics in their multifractal spectrum and deviate substantially from random and uncorrelated sequences of the same size. Small deviations are observed between the longer and the shorter chromosomes, especially for the higher (in absolute values) statistical moments. The correlations are crucial for the form of the multifractal spectrum; surrogate shuffled chromosomes present randomlike spectrum, distinctly different from the actual chromosomes. Analytical approaches based on hierarchical superposition of tensor products show that retaining pair correlations in the sequences leads to a closer representation of the genomic multifractal spectra, especially in the region of negative exponents, due to the underrepresentation of various functional units (such as the cytosine-guanine CG combination and its complementary GC complex). Retaining higher-order correlations in the construction of the tensor products is a way to approach closer the structure of the multifractal spectra of the actual genomic sequences. This hierarchical approach is generic and is applicable to other correlated symbolic sequences.

Long range clustering of oligonucleotides containing the CG signal.

The distance distributions between successive occurrences of the same oligonucleotides in chromosomal DNA are studied, in different classes of higher eucaryotic organisms. A two-parameter modeling is undertaken and applied on the distance distribution of quintuplets (sequences of size five bps) and hexaplets (sequences of size six bps); the first parameter k refers to the short range exponential decay of the distributions, whereas the second parameter m refers to the power law behavior. A two-dimensional scatter plot representing the model equation demonstrates that the points corresponding to the distance distribution of oligonucleotides containing the CG consensus sequence (promoter of the RNA polymerase II) cluster together (group alpha), apart from all other oligonucleotides (group beta). This is shown for the available chordata Homo sapiens, Pan troglodytes, Mus musculus, Rattus norvegicus, Gallus gallus and Danio rerio. This clustering is less evident in lower Animalia and plants, such as Drosophila melanogaster, Caenorhabditis elegans and Arabidopsis thaliana. Moreover, in all organisms the oligonucleotides which contain any consensus sequence are found to be described by long range distributions, whereas all others have a stronger influence of short range decay. Various measures are introduced and evaluated, to numerically characterize the clustering of the two groups. The one which most clearly discriminates the two classes is shown to be the proximity factor.