Microscopic models of traveling wave equations

Reaction-diffusion problems are often described at a macroscopic scale by partial derivative equations of the type of the Fisher or Kolmogorov-Petrovsky-Piscounov equation. These equations have a continuous family of front solutions, each of them corresponding to a different velocity of the front. By simulating systems of size up to N=10^(16) particles at the microscopic scale, where particles react and diffuse according to some stochastic rules, we show that a single velocity is selected for the front. This velocity converges logarithmically to the solution of the F-KPP equation with minimal velocity when the number N of particles increases. A simple calculation of the effect introduced by the cutoff due to the microscopic scale allows one to understand the origin of the logarithmic correction.Comment: 11 pages, 3 figure

An exactly solvable travelling wave equation in the Fisher-KPP class

For a simple one dimensional lattice version of a travelling wave equation, we obtain an exact relation between the initial condition and the position of the front at any later time. This exact relation takes the form of an inverse problem: given the times $t_n$ at which the travelling wave reaches the positions $n$, one can deduce the initial profile. We show, by means of complex analysis, that a number of known properties of travelling wave equations in the Fisher-KPP class can be recovered, in particular Bramson's shifts of the positions. We also recover and generalize Ebert-van Saarloos' corrections depending on the initial condition.Comment: For version 2: some typos + clarification of (87

Ground state energy of a non-integer number of particles with delta attractive interactions

We show how to define and calculate the ground state energy of a system of quantum particles with delta attractive interactions when the number of particles n$is non-integer. The question is relevant to obtain the probability distribution of the free energy of a directed polymer in a random medium. When one expands the ground state energy in powers of the interaction, all the coefficients of the perturbation series are polynomials in n, allowing to define the perturbation theory for non-integer n. We develop a procedure to calculate all the cumulants of the free energy of the directed polymer and we give explicit, although complicated, expressions of the first three cumulants.Comment: 11 pages, no figur

An exactly soluble noisy traveling wave equation appearing in the problem of directed polymers in a random medium

We calculate exactly the velocity and diffusion constant of a microscopic stochastic model of $N$ evolving particles which can be described by a noisy traveling wave equation with a noise of order $N^{-1/2}$. Our model can be viewed as the infinite range limit of a directed polymer in random medium with $N$ sites in the transverse direction. Despite some peculiarities of the traveling wave equations in the absence of noise, our exact solution allows us to test the validity of a simple cutoff approximation and to show that, in the weak noise limit, the position of the front can be completely described by the effect of the noise on the first particle.Comment: 5 page

How genealogies are affected by the speed of evolution

In a series of recent works it has been shown that a class of simple models of evolving populations under selection leads to genealogical trees whose statistics are given by the Bolthausen-Sznitman coalescent rather than by the well known Kingman coalescent in the case of neutral evolution. Here we show that when conditioning the genealogies on the speed of evolution, one finds a one parameter family of tree statistics which interpolates between the Bolthausen-Sznitman and Kingman's coalescents. This interpolation can be calculated explicitly for one specific version of the model, the exponential model. Numerical simulations of another version of the model and a phenomenological theory indicate that this one-parameter family of tree statistics could be universal. We compare this tree structure with those appearing in other contexts, in particular in the mean field theory of spin glasses

Highly Variable Rates of Genome Rearrangements between Hemiascomycetous Yeast Lineages

Hemiascomycete yeasts cover an evolutionary span comparable to that of the entire phylum of chordates. Since this group currently contains the largest number of complete genome sequences it presents unique opportunities to understand the evolution of genome organization in eukaryotes. We inferred rates of genome instability on all branches of a phylogenetic tree for 11 species and calculated species-specific rates of genome rearrangements. We characterized all inversion events that occurred within synteny blocks between six representatives of the different lineages. We show that the rates of macro- and microrearrangements of gene order are correlated within individual lineages but are highly variable across different lineages. The most unstable genomes correspond to the pathogenic yeasts Candida albicans and Candida glabrata. Chromosomal maps have been intensively shuffled by numerous interchromosomal rearrangements, even between species that have retained a very high physical fraction of their genomes within small synteny blocks. Despite this intensive reshuffling of gene positions, essential genes, which cluster in low recombination regions in the genome of Saccharomyces cerevisiae, tend to remain syntenic during evolution. This work reveals that the high plasticity of eukaryotic genomes results from rearrangement rates that vary between lineages but also at different evolutionary times of a given lineage

Improvement in real time detection and selectivity of phthalocyanine gas sensors dedicated to oxidizing pollutants evaluation

International audienceA sensor microsystem prototype, using copper phthalocyanine thin film as sensitive layer, and dedicated to ozone evaluation, was developed. The methodology implemented is based on cyclic sensor recalibrations by thermal cleaning of the sensitive membrane, and on pollutant concentration quantification according to the kinetics of sensor response. Results of laboratory experiments for various NO2 and O3 concentrations, in the range of 10–200 ppb, illustrate the selectivity of CuPc sensors towards ozone, obtained by our methodology. We have shown that ozone selectivity is especially improved for short time of exposure (few minutes) and for phthalocyanine layer maintained at low temperature (80 °C). For optimal conditions, our microsystem exhibits a threshold lower than 10 ppb, a resolution lower than 10 ppb, and good reproducibility of measurements. Performances obtained in real urban atmosphere are satisfying to ensure real time evaluation of ozone during several days. Long-term stability and the detection of NO2 by associating chemical filters to our microsystem will be also discussed