Formes mathématiques, formes physiques, formes vivantes

International audienceChaque forme est une interrogation. Les mathématiques y répondent en termes de structures et de symétries, et elles nous offrent un dictionnaire de formes idéales avec lequel décrire notre perception. La physique nous apprend qu'une forme naturelle révèle tout autant les forces et processus qui l'ont engendrée que les propriétés de la matière dans laquelle elle s'incarne. Le vivant ajoute une dimension évolutive et adaptative pour produire des formes fonctionnelles et des organismes autonomes. Chaque forme est une interrogation. Elle renvoie à une question ancestrale: qu'est-ce que c'est? Est-ce dangereux? L'homme a dû savoir distinguer la forme d'un lion ou celle d'une biche dans une tache beige lointaine. En effet, c'est d'abord et surtout par leur forme que nous identifions les objets qui nous environnent. Celle présentée Figure~1 interpelle ainsi l'observateur, si on ne l'accompagne pas d'explications. Une première réponse à la question de l'identification d'une forme est apportée par les mathématiques. Les objets mathématiques fournissent un répertoire de formes idéales avec lequel décrire les formes réelles de façon plus ou moins approchée. Par exemple, un cercle et son intérieur, le disque, résument les propriétés géométriques communes à tous les ronds: roue, couronne de pain, rond de serviette, globule rouge. Les formes mathématiques les plus typiques, par exemple le cercle ou le carré, sont associées à des symétries remarquables. La formulation opératoire de la symétrie d'une forme est son invariance par un ensemble de transformations: le cercle est invariant par toute rotation autour de son centre, alors qu'un carré est seulement invariant par rotation d'un ou plusieurs quarts de tour autour de son point central, ou bien par réflexion par rapport à un axe médian. Des symétries plus complexes peuvent également se rencontrer, comme par exemple la symétrie hélicoïdale d'une double hélice (forme mathématique de la molécule d'ADN) ou l'autosimilarité d'une structure fractale (forme mathématique d'un relief côtier ou d'un flocon de neige, invariante sous l'action itérée d'un zoom local). En général une forme mathématique peut aussi s'exprimer par une équation. Les formes mathématiques présentent ainsi un aspect géométrique (les symétries qu'elles possèdent). Un cercle peut ainsi être défini par l'équation 'rayon = constante', et il est caractérisé (avec le disque et la couronne) par son invariance par toute rotation autour de son centre. Ces formes mathématiques nous permettent de classifier les formes réelles, de percevoir et de décrire les points communs de leur géométrie, en les associant à des archétypes. Ce sont ces formes mathématiques qui sous-tendent l'esquisse d'un objet réel, lorsqu'on se concentre sur sa forme. Elles réduisent la diversité des formes naturelles à un nombre plus restreint de formes abstraites et idéales. En ce sens les formes naturelles apparaissent comme les avatars multiples des formes idéales rencontrées en mathématiques

How does the chromatin fiber deal with topological constraints?

In the nuclei of eukaryotic cells, DNA is packaged through several levels of compaction in an orderly retrievable way that enables the correct regulation of gene expression. The functional dynamics of this assembly involves the unwinding of the so-called 30 nm chromatin fiber and accordingly imposes strong topological constraints. We present a general method for computing both the twist and the writhe of any winding pattern. An explicit derivation is implemented for the chromatin fiber which provides the linking number of DNA in eukaryotic chromosomes. We show that there exists one and only one unwinding path which satisfies both topological and mechanical constraints that DNA has to deal with during condensation/decondensation processes.Comment: Presented in Nature "News and views in brief" Vol. 429 (13 May 2004). Movies available at http://www.lptl.jussieu.fr/recherche/operationE_fichiers/Page_figurePRL.htm

Intra- and inter-chromosomal interactions correlate with CTCF binding genome wide.

A prime goal in systems biology is the comprehensive use of existing high-throughput genomic datasets to gain a better understanding of chromatin organization and genome function. In this report, we use chromatin immunoprecipitation (ChIP) data that map protein-binding sites on the genome, and Hi-C data that map interactions between DNA fragments in the genome in an integrative approach. We first reanalyzed the contact map of the human genome as determined with Hi-C and found that long-range interactions are highly nonrandom; the same DNA fragments are often found interacting together. We then show using ChIP data that these interactions can be explained by the action of the CCCTC-binding factor (CTCF). These CTCF-mediated interactions are found both within chromosomes and in between different chromosomes. This makes CTCF a major organizer of both the structure of the chromosomal fiber within each individual chromosome and of the chromosome territories within the cell nucleus

Tubulin Dimers Oligomerize before Their Incorporation into Microtubules

In the presence of GTP, purified dimers of α- and β-tubulin will interact longitudinally and laterally to self-assemble into microtubules (MTs). This property provides a powerful in vitro experimental system to describe MT dynamic behavior at the micrometer scale and to study effects and functioning of a large variety of microtubule associated proteins (MAPs). Despite the plethora of such data produced, the molecular mechanisms of MT assembly remain disputed. Electron microscopy (EM) studies suggested that tubulin dimers interact longitudinally to form short oligomers which form a tube by lateral interaction and which contribute to MT elongation. This idea is however challenged: Based on estimated association constants it was proposed that single dimers represent the major fraction of free tubulin. This view was recently supported by measurements suggesting that MTs elongate by addition of single tubulin dimers. To solve this discrepancy, we performed a direct measurement of the longitudinal interaction energy for tubulin dimers. We quantified the size distribution of tubulin oligomers using EM and fluorescence correlation spectroscopy (FCS). From the distribution we derived the longitudinal interaction energy in the presence of GDP and the non-hydrolysable GTP analog GMPCPP. Our data suggest that MT elongation and nucleation involves interactions of short tubulin oligomers rather than dimers. Our approach provides a solid experimental framework to better understand the role of MAPs in MT nucleation and growth

A Physical Model for the Condensation and Decondensation of Eukaryotic Chromosomes

During the eukaryotic cell cycle, chromatin undergoes several conformational changes, which are believed to play key roles in gene expression regulation during interphase, and in genome replication and division during mitosis. In this paper, we propose a scenario for chromatin structural reorganization during mitosis, which bridges all the different scales involved in chromatin architecture, from nucleosomes to chromatin loops. We build a model for chromatin, based on available data, taking into account both physical and topological constraints DNA has to deal with. Our results suggest that the mitotic chromosome condensation/decondensation process is induced by a structural change at the level of the nucleosome itself

An All-Atom Model of the Chromatin Fiber Containing Linker Histones Reveals a Versatile Structure Tuned by the Nucleosomal Repeat Length

In the nucleus of eukaryotic cells, histone proteins organize the linear genome into a functional and hierarchical architecture. In this paper, we use the crystal structures of the nucleosome core particle, B-DNA and the globular domain of H5 linker histone to build the first all-atom model of compact chromatin fibers. In this 3D jigsaw puzzle, DNA bending is achieved by solving an inverse kinematics problem. Our model is based on recent electron microscopy measurements of reconstituted fiber dimensions. Strikingly, we find that the chromatin fiber containing linker histones is a polymorphic structure. We show that different fiber conformations are obtained by tuning the linker histone orientation at the nucleosomes entry/exit according to the nucleosomal repeat length. We propose that the observed in vivo quantization of nucleosomal repeat length could reflect nature's ability to use the DNA molecule's helical geometry in order to give chromatin versatile topological and mechanical properties

Synthetic chromosome fusion: Effects on mitotic and meiotic genome structure and function

We designed and synthesized synI, which is ~21.6% shorter than native chrI, the smallest chromosome in Saccharomyces cerevisiae. SynI was designed for attachment to another synthetic chromosome due to concerns surrounding potential instability and karyotype imbalance and is now attached to synIII, yielding the first synthetic yeast fusion chromosome. Additional fusion chromosomes were constructed to study nuclear function. ChrIII-I and chrIX-III-I fusion chromosomes have twisted structures, which depend on silencing protein Sir3. As a smaller chromosome, chrI also faces special challenges in assuring meiotic crossovers required for efficient homolog disjunction. Centromere deletions into fusion chromosomes revealed opposing effects of core centromeres and pericentromeres in modulating deposition of the crossover-promoting protein Red1. These effects extend over 100 kb and promote disproportionate Red1 enrichment, and thus crossover potential, on small chromosomes like chrI. These findings reveal the power of synthetic genomics to uncover new biology and deconvolute complex biological systems  </p

Nucleosome Chiral Transition under Positive Torsional Stress in Single Chromatin Fibers

Using magnetic tweezers to investigate the mechanical response of single chromatin fibers, we show that fibers submitted to large positive torsion transiently trap positive turns, at a rate of one turn per nucleosome. A comparison with the response of fibers of tetrasomes (the (H3-H4)2 tetramer bound with ~50 bp of DNA) obtained by depletion of H2A-H2B dimers, suggests that the trapping reflects a nucleosome chiral transition to a metastable form built on the previously documented righthanded tetrasome. In view of its low energy, <8 kT, we propose this transition is physiologically relevant and serves to break the docking of the dimers on the tetramer which in the absence of other factors exerts a strong block against elongation of transcription by the main RNA polymerase.Comment: 33 pages (double spacing), 7 figure