Many cellular and developmental processes are tightly controlled by regulatory logic that is similar across different species. Is this similarity the result of common ancestry or is it due to convergent evolution?
We address this question using in silico modeling of genetic networks controlling organ development in flowers. First, we determine the number of networks implementing a given logic and point to an open mathematical problem. Second, we take a computational approach based on Markov Chain Monte Carlo and sample uniformity that highly constrained space of networks. The production of a large number of samples in that ensemble reveals how structural features of genetic networks are shaped by the imposed regulatory logic. Thus, network structural features are expected to be shared across different species through convergent evolution.

The substrate for heredity, DNA, is chemically rather inert. However, it bears one of the elements of information that specify the form of the organism. How can a form be specified, starting from DNA ? Recent observations indicate that the dynamics of transcription — the process that decodes the hereditary information — can imprint forms of a certain topological class onto DNA. This topology allows both to optimize transcription and to facilitate the concerted change of the transcriptional status in response to environmental modifications. To the best of our knowledge, this morphogenetic event is first on the path from DNA to organism. It inspires new rules to optimize networks of transcriptional interactions 'à la manière de' synthetic biology. The latter new interdisciplinary domain will be briefly presented.

It is found a change of variables transforming A2 elliptic one-parametric Hamiltonian (in other words, 3-body elliptic Calogero Hamiltonian, or two-dimensional Lame operator) into algebraic differential operator with polynomial coefficients. It is shown the A2 elliptic model is equivalent to gl(3) quantum top and its parameter is determined by spin of gl(3) representation. It is found the discrete values of the parameter for which two-dimensional Lame operator has polynomial eigenfunctions.

We investigate how deeply nested are the loops in the O(n) model on random maps. In particular, we find that the number P of loops separating two points in a planar map in the dense phase with V >> 1 vertices is typically of order c(n) ln V for a universal constant c(n), and we compute the large deviations of P. The formula we obtain shows similarity to the CLE_{kappa} nesting properties for n = 2cospi(1 – 4/kappa). The results can be extended to all topologies using the topological recursion.

This is based on a joint ongoing work with J. Bouttier

 

Locally analytic representations of p-adic Lie groups are of interest in several branches of arithmetic algebraic geometry, notably the p-adic local Langlands program. I will discuss some work in progress towards a Beilinson-Bernstein style localisation theorem for admissible locally analytic representations of semisimple compact p-adic Lie groups using equivariant formal models of rigid analytic flag varieties.

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The four billion years old genetic code is quasi-universal among living organisms. Mathematical insights into the genetic code can be gained by studying its symmetries. Symmetries by base substitutions were identified for the 2' and 3' sets of aminoacyl-tRNAs with Soulé in 2007. Half a century ago, Rumer described symmetries by base substitutions for degeneracy, the number of base triplets coding for each of the twenty canonical amino acids found in natural proteins. Apart from base substitutions, another class of mutations consists of frameshift mutations. The most frequent frameshift mutations are single-base deletions which are highly deleterious within genes as they alter the reading frame. The central role of mutations which are errors induced by biological catalysts, will be analyzed in the context of the minimization of their deleterious effects, the codon assignment of amino acids and chain termination signals in the genetic code and molecular evolution.
 

The central axis of the famous DNA double helix is often constrained or even circular.   The topology of this axis can influence which proteins interact with the underlying DNA. Subsequently, in all cells there are proteins whose primary function (type II topoisomerases)  is to change the DNA axis topology — for example converting a torus link into an unknot. Additionally, there are several protein families (most importantly, site-specific recombinases) that change the axis topology as a by-product of their interaction with DNA.

This talk will describe some typical DNA conformations, and the families of proteins that change these conformations. I'll present a few examples illustrating how 3-manifold topology (including Dehn surgery and Heegaard Floer homology) have been useful in understanding certain DNA-protein interactions, and discuss the most common techniques used to attack these problems.

Une des définitions possibles des nombres de Hurwitz est qu'ils comptent certaines cartes étiquetées. De même que les cartes planaires sont en bijection avec des arbres plans (ou ordonnés), ces cartes étiquetées sont en bijection avec des arbres étiquetés de type Cayley. Le but de l'exposé sera de montrer comment l'étude de ces arbres permet de d'obtenir de nouvelles formules pour les nombres de Hurwitz doubles ainsi que des propriétés de polynomialité.

Considérons une marche aléatoire simple sur un arbre de Galton-Watson critique conditionné à avoir une hauteur supérieure à $n$. La loi du point d'atteinte de la hauteur $n$ par la marche aléatoire s'appelle la mesure harmonique au niveau $n$. Il est bien connu que le cardinal de l'ensemble des sommets de l'arbre au niveau $n$ est de l'ordre de $n$. En 2013 Curien et Le Gall ont prouvé qu'il existe une constante $beta=0.78…$ telle que la mesure harmonique est portée, à un ensemble de masse arbitrairement petite près, par un ensemble de cardinal de l'ordre de $n^beta$. Dans cet exposé, nous présentons l'existence d'une nouvelle constante universelle $lambda=1.21…$ telle que, avec grande probabilité, la mesure harmonique portée par un sommet typique à la hauteur $n$ est de l'ordre $n^{-lambda}$.