Nikolaos Bournaveas

Directional persistence of chemotactic bacteria in a traveling concentration wave

Chemotactic bacteria are known to collectively migrate towards sources of attractants. In confined convectionless geometries, concentration “waves” of swimming Escherichia coli can form and propagate through a self-organized process involving hundreds of thousands of these microorganisms. These waves are observed in particular in microcapillaries or microchannels; they result from the interaction between individual chemotactic bacteria and the macroscopic chemical gradients dynamically generated by the migrating population. By studying individual trajectories within the propagating wave, we show that, not only the mean run length is longer in the direction of propagation, but also that the directional persistence is larger compared to the opposite direction. This modulation of the reorientations significantly improves the efficiency of the collective migration. Moreover, these two quantities are spatially modulated along the concentration profile. We recover quantitatively these microscopic and macroscopic observations with a dedicated kinetic model.

Mathematical Description of Bacterial Traveling Pulses

The Keller-Segel system has been widely proposed as a model for bacterial waves driven by chemotactic processes. Current experiments on Escherichia coli have shown the precise structure of traveling pulses. We present here an alternative mathematical description of traveling pulses at the macroscopic scale. This modeling task is complemented with numerical simulations in accordance with the experimental observations. Our model is derived from an accurate kinetic description of the mesoscopic run-and-tumble process performed by bacteria. This can account for recent experimental observations with E. coli. Qualitative agreements include the asymmetry of the pulse and transition in the collective behaviour (clustered motion versus dispersion). In addition, we can capture quantitatively the traveling speed of the pulse as well as its characteristic length. This work opens several experimental and theoretical perspectives since coefficients at the macroscopic level are derived from considerations at the cellular scale. For instance, the particular response of a single cell to chemical cues turns out to have a strong effect on collective motion. Furthermore, the bottom-up scaling allows us to perform preliminary mathematical analysis and write efficient numerical schemes. This model is intended as a predictive tool for the investigation of bacterial collective motion.

Local existence for the maxwell-dirac equations in three space dimensions

In this paper we study the initial value problem for the Maxwell-Dirac equations in 3+1-dimensional Minkowski spacetime. The linear Dirac equation is the Euler-Lagrange equation corresponding to the Lagrangian. Here {psi} and {psi} are formally regarded as being independent fields. If we vary {psi} we get the Dirac equation for {psi}. By varying {psi} we get the conjugate Dirac equation for {psi}. We have used the following notation: {psi} denotes a 4-spinor field defined on R{sup 3+1}. It is represented as a column vector with 4 components. {psi} denotes the complex conjugate transpose of {psi}; it is a row vector with 4 components. 14 refs.

The one-dimensional Keller–Segel model with fractional diffusion of cells

We investigate the one-dimensional Keller–Segel model where the diffusion is replaced by a non-local operator, namely the fractional diffusion with exponent 0 < α ≤ 2. We prove some features related to the classical two-dimensional Keller–Segel system: blow-up may or may not occur depending on the initial data. More precisely a singularity appears in finite time when α < 1 and the initial configuration of cells is sufficiently concentrated. On the other hand, global existence holds true for α ≤ 1 if the initial density is small enough in the sense of the L1/α norm.

Global Existence for a Kinetic Model of Chemotaxis via Dispersion and Strichartz Estimates

We investigate further the existence of solutions to kinetic models of chemotaxis. These are nonlinear transport-scattering equations with a quadratic nonlinearity which have been used to describe the motion of bacteria since the 80s when experimental observations have shown they move by a series of ‘run and tumble’. The existence of solutions has been obtained in several papers Chalub et al. (2004), Hwang et al. (2005a b) using direct and strong dispersive effects. Here, we use the weak dispersion estimates of Castella and Perthame (1996) to prove global existence in various situations depending on the turning kernel. In the most difficult cases, where both the velocities before and after tumbling appear, with the known methods, only Strichartz estimates can give a result, with a smallness assumption.

LOW REGULARITY SOLUTIONS OF THE DIRAC KLEIN-GORDON EQUATIONS IN TWO SPACE DIMENSIONS

In this paper we study low regularity solutions of the Dirac KleinGordon equations in the case of two space dimensions. This system, together with the Maxwell–Dirac equations, form the foundations of Relativistic Electrodynamics [2, 3, 4]. Our motivation was the work [16] of Y.X.Zheng on a modified version of the system. Let ðt, xÞ be a real valued scalar field and ðt, xÞ be a 2-spinor field. The Dirac operator D is defined by D 1⁄4 i @ where , 1⁄4 0, 1, 2 are the Dirac matrices. The wave operator & is defined by & 1⁄4 @t þ . Then D 1⁄4 &. We denote by y the complex conjugate transpose of and we define 1⁄4 y . The Dirac Klein-Gordon equations are then written as: D 1⁄4 g M ð1Þ & 1⁄4 g þm ð2Þ

Averages over spheres for kinetic transport equations; hyperbolic Sobolev spaces and Strichartz inequalities

We consider averages over spheres for kinetic transport equations in two space dimensions. In this case, 1/4 derivative is lost in the various forms of the averaging lemmas. We show that it is possible to recover the optimal regularity working in the hyperbolic Sobolev spaces. Strichartz type inequalities follow with better exponents than those given by classical Sobolev imbeddings.

Critical mass phenomenon for a chemotaxis kinetic model with spherically symmetric initial data

Abstract The goal of this paper is to exhibit a critical mass phenomenon occurring in a model for cell self-organization via chemotaxis. The very well-known dichotomy arising in the behavior of the macroscopic Keller–Segel system is derived at the kinetic level, being closer to microscopic features. Indeed, under the assumption of spherical symmetry, we prove that solutions with initial data of large mass blow-up in finite time, whereas solutions with initial data of small mass do not. Blow-up is the consequence of a momentum computation and the existence part is derived from a comparison argument. Spherical symmetry is crucial within the two approaches. We also briefly investigate the drift-diffusion limit of such a kinetic model. We recover partially at the limit the Keller–Segel criterion for blow-up, thus arguing in favour of a global link between the two models.

Local well-posedness for the space–time Monopole equation in Lorenz gauge

It is known from Czubak (Anal PDE 3(2):151–174, 2010) that the space–time Monopole equation is locally well-posed in the Coulomb gauge for small initial data in