Stability in Hamiltonian systems : applications to the restricted three-body problem

The N-body problem is a classical famous problem which has attracted a lot of attention. It consists of describing the complete behavior of all solutions of the equations of motions for a given initial condition. Still related to this kind of problem Euler in 1772 describe the three-body problem in his eort to study the motion of the moon. Later on Jacobi in 1836 brought forward the main mathematical interest in an even more specic part of the three body problem, namely the one which is reduced to a conservative two degrees of freedom problem. This has somehow brought up an extensive study on mechanics. Despite of all this eort of great mathematicians, in general the N-body problem for N> 2 is still unsolved

A Dynamically Diluted Alignment Model Reveals the Impact of Cell Turnover on the Plasticity of Tissue Polarity Patterns

The polarisation of cells and tissues is fundamental for tissue morphogenesis during biological development and regeneration. A deeper understanding of biological polarity pattern formation can be gained from the consideration of pattern reorganisation in response to an opposing instructive cue, which we here consider by example of experimentally inducible body axis inversions in planarian flatworms. Our dynamically diluted alignment model represents three processes: entrainment of cell polarity by a global signal, local cell-cell coupling aligning polarity among neighbours and cell turnover inserting initially unpolarised cells. We show that a persistent global orienting signal determines the final mean polarity orientation in this stochastic model. Combining numerical and analytical approaches, we find that neighbour coupling retards polarity pattern reorganisation, whereas cell turnover accelerates it. We derive a formula for an effective neighbour coupling strength integrating both effects and find that the time of polarity reorganisation depends linearly on this effective parameter and no abrupt transitions are observed. This allows to determine neighbour coupling strengths from experimental observations. Our model is related to a dynamic $8$-Potts model with annealed site-dilution and makes testable predictions regarding the polarisation of dynamic systems, such as the planarian epithelium.Comment: Preprint as prior to first submission to Journal of the Royal Society Interface. 25 pages, 6 figures, plus supplement (18 pages, contains 1 table and 7 figures). A supplementary movie is available from https://dx.doi.org/10.6084/m9.figshare.c388781

Multicellularity in animals: The potential for within-organism conflict.

Metazoans function as individual organisms but also as “colonies” of cells whose single-celled ancestors lived and reproduced independently. Insights from evolutionary biology about multicellular group formation help us understand the behavior of cells: why they cooperate, and why cooperation sometimes breaks down. Current explanations for multicellularity focus on two aspects of development which promote cooperation and limit conflict among cells: a single-cell bottleneck, which creates organisms composed of clones, and a separation of somatic and germ cell lineages, which reduces the selective advantage of cheating. However, many obligately multicellular organisms thrive with neither, creating the potential for within-organism conflict. Here, we argue that the prevalence of such organisms throughout the Metazoa requires us to refine our preconceptions of conflict-free multicellularity. Evolutionary theory must incorporate developmental mechanisms across a broad range of organisms—such as unusual reproductive strategies, totipotency, and cell competition—while developmental biology must incorporate evolutionary principles. To facilitate this cross-disciplinary approach, we provide a conceptual overview from evolutionary biology for developmental biologists, using analogous examples in the well-studied social insects

Kinetic Characterization and X-ray Structure of a Mutant of Haloalkane Dehalogenase with Higher Catalytic Activity and Modified Substrate Range

Conversion of halogenated aliphatics by haloalkane dehalogenase proceeds via the formation of a covalent alkyl-enzyme intermediate which is subsequently hydrolyzed by water. In the wild type enzyme, the slowest step for both 1,2-dichloroethane and 1,2-dibromoethane conversion is a unimolecular enzyme isomerization preceding rapid halide dissociation. Phenylalanine 172 is located in a helix-loop-helix structure that covers the active site cavity of the enzyme, interacts with the Clβ of 1,2-dichloroethane during catalysis, and could be involved in stabilization of this helix-loop-helix region of the cap domain of the enzyme. To obtain more information about the role of this residue in dehalogenase function, we performed a mutational analysis of position 172 and studied the kinetics and X-ray structure of the Phe172Trp enzyme. The Phe172Trp mutant had a 10-fold higher kcat/Km for 1-chlorohexane and a 2-fold higher kcat for 1,2-dibromoethane than the wild-type enzyme. The X-ray structure of the Phe172Trp enzyme showed a local conformational change in the helix-loop-helix region that covers the active site. This could explain the elevated activity for 1-chlorohexane of the Phe172Trp enzyme, since it allows this large substrate to bind more easily in the active site cavity. Pre-steady-state kinetic analysis showed that the increase in kcat found for 1,2-dibromoethane conversion could be attributed to an increase in the rate of an enzyme isomerization step that preceeds halide release. The observed conformational difference between the helix-loop-helix structures of the wild-type enzyme and the faster mutant suggests that the isomerization required for halide release could be a conformational change that takes place in this region of the cap domain of the dehalogenase. It is proposed that Phe172 is involved in stabilization of the helix-loop-helix structure that covers the active site of the enzyme and creates a rigid hydrophobic cavity for small apolar halogenated alkanes.

Vortices on Hyperbolic Surfaces

It is shown that abelian Higgs vortices on a hyperbolic surface $M$ can be constructed geometrically from holomorphic maps $f:M \to N$, where $N$ is also a hyperbolic surface. The fields depend on $f$ and on the metrics of $M$ and $N$. The vortex centres are the ramification points, where the derivative of $f$ vanishes. The magnitude of the Higgs field measures the extent to which $f$ is locally an isometry. Witten's construction of vortices on the hyperbolic plane is rederived, and new examples of vortices on compact surfaces and on hyperbolic surfaces of revolution are obtained. The interpretation of these solutions as SO(3)-invariant, self-dual SU(2) Yang--Mills fields on $\R^4$ is also given.Comment: Revised version: new section on four-dimensional interpretation of hyperbolic vortices added

Kolmogorov condition near hyperbolic singularities of integrable Hamiltonian systems

In this paper we show that, if an integrable Hamiltonian system admits a nondegenerate hyperbolic singularity then it will satisfy the Kolmogorov condegeneracy condition near that singularity (under a mild additional condition, which is trivial if the singularity contains a fixed point)Comment: revised version, 11p, accepted for publication in a sepecial volume in Regular and Chaotic Dynamics in honor of Richard Cushma

Measurement of miniband parameters of a doped superlattice by photoluminescence in high magnetic fields

We have studied a 50/50\AA superlattice of GaAs/Al$_{0.21}$Ga$_{0.79}$As composition, modulation-doped with Si, to produce $n=1.4\times 10^{12}$ cm$^{-2}$ electrons per superlattice period. The modulation-doping was tailored to avoid the formation of Tamm states, and photoluminescence due to interband transitions from extended superlattice states was detected. By studying the effects of a quantizing magnetic field on the superlattice photoluminescence, the miniband energy width, the reduced effective mass of the electron-hole pair, and the band gap renormalization could be deduced.Comment: minor typing errors (minus sign in eq. (5)

Ultrafast pump-probe dynamics in ZnSe-based semiconductor quantum-wells

Pump-probe experiments are used as a controllable way to investigate the properties of photoexcited semiconductors, in particular, the absorption saturation. We present an experiment-theory comparison for ZnSe quantum wells, investigating the energy renormalization and bleaching of the excitonic resonances. Experiments were performed with spin-selective excitation and above-bandgap pumping. The model, based on the semiconductor Bloch equations in the screened Hartree-Fock approximation, takes various scattering processes into account phenomenologically. Comparing numerical results with available experimental data, we explain the experimental results and find that the electron spin-flip occurs on a time scale of 30 ps.Comment: 10 pages, 9 figures. Key words: nonlinear and ultrafast optics, modeling of femtosecond pump-probe experiments, electron spin-flip tim