Laurent Vuillon

From local to global changes in proteins: a network view.

To fulfill the biological activities in living organisms, proteins are endowed with dynamics, robustness and adaptability. The three properties co-exist because they allow global changes in structure to arise from local perturbations (dynamics). Robustness refers to the ability of the protein to incur such changes without suffering loss of function; adaptability is the emergence of a new biological activity. Since loss of function may jeopardize the survival of the organism and lead to disease, adaptability may occur through the combination of two local perturbations that together rescue the initial function. The review highlights the relevancy of computational network analysis to understand how a local change produces global changes.

Asymptotic behavior in a heap model with two pieces

In a heap model, solid blocks, or pieces, pile up according to the Tetris game mechanism. An optimal schedule is an infinite sequence of pieces minimizing the asymptotic growth rate of the heap. In a heap model with two pieces, we prove that there always exists an optimal schedule which is balanced, either periodic or Sturmian. We also consider the model where the successive pieces are chosen at random, independently and with some given probabilities. We study the expected growth rate of the heap. For a model with two pieces, the rate is either computed explicitly or given as an infinite series. We show an application for a system of two processes sharing a resource, and we prove that a greedy schedule is not always optimal.

Palindromic language of thin discrete planes

We work on the Rveills hyperplane P(v,0,) with normal vectorvRd, shift=0 and thicknessR. Such a hyperplane is connected as soon as is greater than some value (v,0), called the connecting thickness of v with null shift. In the case where v satisfies the so called Kraaikamp and Meester criterion, at the connecting thickness the hyperplane has very specific properties. First of all the adjacency graph of the voxels forms a tree. This tree appeared in many works both in discrete geometry and in discrete dynamical systems. In addition, it is well known that for a finite coding of length n of discrete lines, the number of palindromes in the language is exactly n+1. We extend this notion of language to labeled trees and we compute the number of distinct palindromes. In fact for our voxel adjacency trees with n letters we show that the number of palindromes in the language is also n+1. This result establishes a first link between combinatorics on words, palindromic languages, voxel adjacency trees and connecting thickness of Rveills hyperplanes. It also provides a better understanding of the combinatorial structure of discrete planes.

Some special solutions to the Hyperbolic NLS equation

The Hyperbolic Nonlinear Schrodinger equation (HypNLS) arises as a model for the dynamics of three-dimensional narrowband deep water gravity waves. In this study, the Petviashvili method is exploited to numerically compute bi-periodic time-harmonic solutions of the HypNLS equation. In physical space they represent non-localized standing waves. Non-trivial spatial patterns are revealed and an attempt is made to describe them using symbolic dynamics and the language of substitutions. Finally, the dynamics of a slightly perturbed standing wave is numerically investigated by means a highly acccurate Fourier solver.