Scale-free duplication dynamics: a model for ultraduplication

Analysis of human DNA through power-law statistics

We report an analysis of Homo sapiens DNA through the formalism of κ statistics, which encompasses power-law correlations and provides an optimization principle that permits us to model distinct physical systems; i.e., the power-law distribution of the length of DNA bases is calculated from a general model which follows arguments similar to those proposed in Maxwell's deduction of statistical distributions. The viability of the model is tested using a data set from a catalog of proteins collected from the Ensembl Project. The results indicate that the short-range correlations, always present in coding DNA sequences, are appropriately captured through the Kaniadakis power-law distribution, adequately describing the cumulative length distribution of DNA bases, in contrast with the case of the traditional exponential statistical model.

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.

Size distribution of function-based human gene sets and the split-merge model

The sizes of paralogues-gene families produced by ancestral duplication-are known to follow a power-law distribution. We examine the size distribution of gene sets or gene families where genes are grouped by a similar function or share a common property. The size distribution of Human Gene Nomenclature Committee (HGNC) gene sets deviate from the power-law, and can be fitted much better by a beta rank function. We propose a simple mechanism to break a power-law size distribution by a combination of splitting and merging operations. The largest gene sets are split into two to account for the subfunctional categories, and a small proportion of other gene sets are merged into larger sets as new common themes might be realized. These operations are not uncommon for a curator of gene sets. A simulation shows that iteration of these operations changes the size distribution of Ensembl paralogues and could lead to a distribution fitted by a rank beta function. We further illustrate application of beta rank function by the example of distribution of transcription factors and drug target genes among HGNC gene families.

Ising-model description of long-range correlations in DNA sequences

We model long-range correlations of nucleotides in the human DNA sequence using the long-range one-dimensional (1D) Ising model. We show that, for distances between 10(3) and 10(6) bp, the correlations show a universal behavior and may be described by the non-mean-field limit of the long-range 1D Ising model. This allows us to make some testable hypothesis on the nature of the interaction between distant portions of the DNA chain which led to the DNA structure that we observe today in higher eukaryotes.

Exhaustive computation of exact duplications via super and non-nested local maximal repeats

We propose and implement a method to obtain all duplicated sequences (repeats) from a chromosome or whole genome. Unlike existing approaches our method makes it possible to simultaneously identify and classify repeats into super, local, and non-nested local maximal repeats. Computation verification demonstrates that maximal repeats for a genome of several gigabases can be identified in a reasonable time, enabling us to identified these maximal repeats for any sequenced genome. The algorithm used for the identification relies on enhanced suffix array data structure to achieve practical space and time efficiency, to identify and classify the maximal repeats, and to perform further post-processing on the identified duplicated sequences. The simplicity and effectiveness of the implementation makes the method readily extendible to more sophisticated computations. Maxmers can be exhaustively accounted for in few minutes for genome sequences of dozen megabases in length and in less than a day or two for genome sequences of few gigabases in length. One application of duplicated sequence identification is to the study of duplicated sequence length distributions, which our found to exhibit for large lengths a persistent power-law behavior. Variation of estimated exponents of this power law are studied among different species and successive assembly release versions of the same species. This makes the characterization of the power-law regime of sequenced genomes via maximal repeats identification and classification, an important task for the derivation of models that would help us to elucidate sequence duplication and genome evolution.

Neutral evolution of duplicated DNA: an evolutionary stick-breaking process causes scale-invariant behavior

Recently, an enrichment of identical matching sequences has been found in many eukaryotic genomes. Their length distribution exhibits a power law tail raising the question of what evolutionary mechanism or functional constraints would be able to shape this distribution. Here we introduce a simple and evolutionarily neutral model, which involves only point mutations and segmental duplications, and produces the same statistical features as observed for genomic data. Further, we extend a mathematical model for random stick breaking to analytically show that the exponent of the power law tail is -3 and universal as it does not depend on the microscopic details of the model.