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Fractals and Wavelets : What Can We Learn on Transcription and Replication from Wavelet-Based Multifractal Analysis of DNA Sequences ?

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Mathematics of Complexity and Dynamical Systems

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Article Outline

Glossary

Definition of the Subject

Introduction

A Wavelet-Based Multifractal Formalism: The Wavelet Transform Modulus Maxima Method

Bifractality of Human DNA Strand-Asymmetry Profiles Results from Transcription

From the Detection of Relication Origins Using the Wavelet Transform Microscope to the Modeling of Replication in Mammalian Genomes

A Wavelet-Based Methodology to Disentangle Transcription- and Replication-Associated Strand Asymmetries Reveals a Remarkable Gene Organization in the Human Genome

Future Directions

Acknowledgments

Bibliography

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Abbreviations

Fractal :

Fractals are complex mathematical objects that are invariant with respect to dilations ( self‐similarity ) and therefore do not possess a characteristic length scale. Fractal objects display scale‐invariance properties that can either fluctuate from point to point ( multifractal ) or be homogeneous ( monofractal ). Mathematically, these properties should hold over all scales. However, in the real world, there are necessarily lower and upper bounds over which self‐similarity applies.

Wavelet transform :

The continuous wavelet transform (WT) is a mathematical technique introduced in the early 1980s to perform time‐frequency analysis. The WT has been early recognized as a mathematical microscope that is well adapted to characterize the scale‐invariance properties of fractal objects and to reveal the hierarchy that governs the spatial distribution of the singularities of multifractal measures and functions. More specifically, the WT is a space-scale analysis which consists in expanding signals in terms of wavelets that are constructed from a single function, the analyzing wavelet, by means of translations and dilations.

Wavelet transform modulus maxima method :

The WTMM method provides a unified statistical (thermodynamic) description of multifractal distributions including measures and functions. This method relies on the computation of partition functions from the wavelet transform skeleton defined by the wavelet transform modulus maxima (WTMM). This skeleton provides an adaptive space-scale partition of the fractal distribution under study, from which one can extract the \({D(h)}\) singularity spectrum as the equivalent of a thermodynamic potential (entropy). With some appropriate choice of the analyzing wavelet, one can show that the WTMM method provides a natural generalization of the classical box‐counting and structure function techniques.

Compositional strand asymmetry :

The DNA double helix is made of two strands that are maintained together by hydrogen bonds involved in the base‐pairing between Adenine (resp. Guanine) on one strand and Thymine (resp. Cytosine) on the other strand. Under no‐strand bias conditions, i. e. when mutation rates are identical on the two strands, in other words when the two strands are strictly equivalent, one expects equimolarities of adenine and thymine and of guanine and cytosine on each DNA strand, a property named Chargaff's second parity rule. Compositional strand asymmetry refers to deviations from this rule which can be assessed by measuring departure from intrastrand equimolarities. Note that two major biological processes, transcription and replication , both requiring the opening of the double helix, actually break the symmetry between the two DNA strands and can thus be at the origin of compositional strand asymmetries.

Eukaryote:

Organisms whose cells contain a nucleus, the structure containing the genetic material arranged into chromosomes. Eukaryotes constitute one of the three domains of life, the two others, called prokaryotes (without nucleus), being the eubacteria and the archaebacteria.

Transcription :

Transcription is the process whereby the DNA sequence of a gene is enzymatically copied into a complementary messenger RNA. In a following step, translation takes place where each messenger RNA serves as a template to the biosynthesis of a specific protein.

Replication :

DNA replication is the process of making an identical copy of a double‐stranded DNA molecule. DNA replication is an essential cellular function responsible for the accurate transmission of genetic information though successive cell generations. This process starts with the binding of initiating proteins to a DNA locus called origin of replication . The recruitment of additional factors initiates the bi‐directional progression of two replication forks along the chromosome. In eukaryotic cells, this binding event happens at a multitude of replication origins along each chromosome from which replication propagates until two converging forks collide at a  terminus of replication .

Chromatin :

Chromatin is the compound of DNA and proteins that forms the chromosomes in living cells. In eukaryotic cells, chromatin is located in the nucleus.

Histones:

Histones are a major family of proteins found in eukaryotic chromatin. The wrap** of DNA around a core of 8 histones forms a  nucleosome , the first step of eukaryotic DNA compaction.

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Acknowledgments

We thank O. Hyrien, F. Mongelard and C. Moskalenko for interesting discussions. This work was supported by the Action Concertée Incitative Informatique, Mathématiques, Physique en Biologie Moléculaire 2004 under the project “ReplicOr”, the Agence Nationale de la Recherche under the project “HUGOREP” and the program “Emergence” of the Conseil Régional Rhônes-Alpes and by the Programme d'Actions Intégrées Tournesol.

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Authors and Affiliations

  1. Laboratoire Joliot–Curie and Laboratoire de Physique, ENS-Lyon CNRS, Lyon Cedex, France

    Alain Arneodo, Benjamin Audit & Edward-Benedict Brodie of Brodie

  2. Institut de Mathématique, Université de Liège, Liège, Belgium

    Samuel Nicolay

  3. Génétique des Génomes Bactériens, Institut Pasteur, CNRS, Paris, France

    Marie Touchon

  4. Centre de Génétique Moléculaire, CNRS, Gif-sur-Yvette, France

    Yves d'Aubenton-Carafa, Maxime Huvet & Claude Thermes

  5. Atelier de Bioinformatique, Université Pierre et Marie Curie, Paris, France

    Marie Touchon

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Arneodo, A. et al. (2012). Fractals and Wavelets : What Can We Learn on Transcription and Replication from Wavelet-Based Multifractal Analysis of DNA Sequences ?. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_39

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