Complex multifractal nature in Mycobacterium tuberculosis genome

Abstract

The mutifractal and long range correlation (C(r)) properties of strings, such as nucleotide sequence can be a useful parameter for identification of underlying patterns and variations. In this study C(r) and multifractal singularity function f(α) have been used to study variations in the genomes of a pathogenic bacteria Mycobacterium tuberculosis. Genomic sequences of M. tuberculosis isolates displayed significant variations in C(r) and f(α) reflecting inherent differences in sequences among isolates. M. tuberculosis isolates can be categorised into different subgroups based on sensitivity to drugs, these are DS (drug sensitive isolates), MDR (multi-drug resistant isolates) and XDR (extremely drug resistant isolates). C(r) follows significantly different scaling rules in different subgroups of isolates, but all the isolates follow one parameter scaling law. The richness in complexity of each subgroup can be quantified by the measures of multifractal parameters displaying a pattern in which XDR isolates have highest value and lowest for drug sensitive isolates. Therefore C(r) and multifractal functions can be useful parameters for analysis of genomic sequences.

Figure 1. Multifractal and correlation function behaviors of rpoB gene (sequence positions: 759807–763325), phoP and phoR gene combined together(sequence positions: 851608 to 853853) in M. tuberculosis genome.

(a) DNA walks of forty isloates each of DS, MDR and XDR of M. tuberculosis (panels of uppermost row). (b) Corresponding plots of correlation functions (C(r) versus r) of the three types of isloates (panels of middle row). Straight lines are power law fits on the data (for rpoB gene: DS: ; MDR: ; XDR: and for phoPR gene complex: DS: ; MDR: ; XDR: ). (c) Plots of fluctuation function () with respect to s for the corresponding three types of M. tuberculosis isloates showing power law nature (panels of lowermost or third row).

Figure 2. Multifractal and correlation function behaviors of gyrB gene and gyrA gene concatenated together (sequence positions: 5240–9810) and embC, embA and embB gene concatenated together with sequence position from 4239863 to 4249810 in M. tuberculosis genome.

(a) DNA walks of forty isloates each of DS, MDR and XDR of M. tuberculosis (panels of uppermost row). (b) Corresponding plots of correlation functions (C(r) versus r) of the three types of isloates (panels of middle row). Straight lines are power law fits on the data (for gyrBA: DS: ; MDR: ; XDR: and for embCAB: DS: ; MDR: ; XDR: ). (c) Plots of fluctuation function () with respect to s for the corresponding three types of M. tuberculosis isloates showing power law nature (panels of lowermost or third row).

Figure 3. Singularity spectrum of rpoB gene and phoPR gene complex of M. tuberculosis isolates (forty) of each DS, MDR and XDR.

(a) Plots of singularity function f(α) of the three types of isolates with respect to α (panels of first row) and their Scaling of f(α) by choosing α_c = 1.02275 using interpolation showing self-affine process of the isolates (inside box). (b) Properties of multifractal spectral parameters: behaviors of α₀, Δα and χ as a function of isolates (colors show types of isolates).

Figure 4. Singularity spectrum of gyrBA gene complex(sequence position: 5240 to 9810) and embCAB gene complex(sequence position: 4239863 to 4249810) of M. tuberculosis isolates (forty) of each DS, MDR and XDR.

(a) Plots of singularity function f(α) of the three types of isolates with respect to α (panels of first row) and their Scaling of f(α) by choosing α_c = 1.02275 using interpolation showing self-affine process of the isolates (inside box). (b) Properties of multifractal spectral parameters: behaviors of α₀, Δα and χ as a function of isolates (colors show types of isolates).

Figure 5. Multifractal and correlation function behaviors of all SNPs (SNPs based Sequences) within a genome of forty isolates each from DS, MDR and XDR.

(a) DNA walks of forty isolates each of DS, MDR and XDR of M. tuberculosis (the first three panels of uppermost row). (b) Corresponding plots of correlation functions (C(r) versus r) of the three types of isolates (first three panels of second row). Straight lines are power law fits on the data (for DS: ; MDR: ; XDR: ). (c) Plots of singularity function f(α) of the three types of isolates with respect to α (first three panels of third row). (d) Properties of multifractal spectral parameters: behaviors of α₀, Δα and χ as a function of isolates (colors show types of isolates) in the bottom row.

Figure 6. One parameter scaling law in correlation function of rpoB gene and SNP based sequences of DS, MDR and XDR of M. tuberculosis.

(a) Scaling in rpoB gene for twenty isolates each of DS, MDR and XDR (panels of first row). The straight lines are power law fits to each isolate. The power law exponent (γ) for DS, MDR and XDR are given in rightmost panel of second row. The scaled data using Mackinnon and Kramer’s one parameter scaling procedure ( as a function of r, see Methods) is shown in first two panels of second row. (b) Same scaling procedure is done for SNP based sequences (panels of third and fourth row).

Figure 7. Multifractal and correlation function based classification of DS, MDR and XDR.

The average singularity spectra of DS, MDR and XDR of rpoB, phoPR, gyrBA, embCAB and SNP based sequences (lower panel) as a function of α. The classification of DS, MDR and XDR based on Multifractal parameters and correlation function behaviors.

Figure 8. Computational Pipeline for Multifractal Analysis in M. tuberculosis bacterium.