A clock and wavefront mechanism for somite formation

Somitogenesis, the sequential formation of a periodic pattern along the antero-posterior axis of vertebrate embryos, is one of the most obvious examples of the segmental patterning processes that take place during embryogenesis and also one of the major unresolved events in developmental biology. In this article, we develop a mathematical formulation of a new version of the Clock and Wavefront model proposed by Pourquié and co-workers (Dubrulle, J., McGrew, M.J., Pourquié, O., 2001. FGF signalling controls somite boundary position and regulates segmentation clock control of spatiotemporal Hox gene activation. Cell 106, 219–232). Dynamic expression of FGF8 in the presomitic mesoderm constitutes the wavefront of determination which sweeps along the body axis interacting as it moves with the segmentation clock to gate cells into somites. We also show that the model can mimic the anomalies formed when progression of the wavefront is disturbed and make some experimental predictions that can be used to test the hypotheses underlying the model

Accurate Prediction of Nonlinear Wave Forces: Part I (Fixed Cylinder)

A new equation structure is proposed as an alternative to the Morison equation for the prediction of wave forces. Initially, nonlinear parametric continuous time differential equation models were estimated from wave force data for a variety of flow situations by adopting a new approach which avoids direct differentiation of the input-output data. The method consists of two stages. The first stage involves estimation of a discrete time model (polynomial NARMAX) from sampled input-output data and computation of the linear and higher order frequency response functions. The second stage involves identifying continuous time models by curve fitting to the complex frequency response data using a weighted complex orthogonal estimator. The orthogonal property of the estimator helps in identifying the correct model structure or which terms to include in the model and the weighting property provides an additional degree of freedom to control the properties of the estimator with respect to the selection of the frequency range and number of data points. Morison equation models were initially fitted to the data but were shown to simply curve fit to the data without capturing the underlying dynamics. The frequency domain characteristics of the Morison equation models were also analysed and shown to be structurally deficient in representing certain dynamic features of the force. However, it is shown that the new equation structure is capable of emulating all the relevant features of the wave force mechanics. The paper is organised in two parts. Part I is concerned with the modelling of wave forces on a fixed cylinder and Part II deals with a responding cylinder. Extensive simulations on a variety of experimental data show that models based on the new structure perform remarkably well compared with the Morison equation. For each flow situation, in addition to the drag and inertia coefficients of the Morison equation, there are two non-dimensional coefficients defining history effects, which show some consistency between widely different flow situations

Accurate Prediction of Nonlinear Wave Forces: Part II(Responding Cylinder)

In part I of this paper, continuous time models have been fitted to wave forces acting on fixed cylinders and a new equation structure called the Dynamic Morison model has been proposed as an alternative to the traditional Morison equation. In this, the second part of the paper, continuous time nonlinear differential equation models are fitted to wave forces from responding cylinders and corresponding equation structures are proposed as an alternative to the Morison equation for the prediction of wave forces for the moving cylinder case. It is shown that the new models explain the wave force mechanisms and perform considerably better than the Morison equation

Virus shapes and buckling transitions in spherical shells

We show that the icosahedral packings of protein capsomeres proposed by Caspar and Klug for spherical viruses become unstable to faceting for sufficiently large virus size, in analogy with the buckling instability of disclinations in two-dimensional crystals. Our model, based on the nonlinear physics of thin elastic shells, produces excellent one parameter fits in real space to the full three-dimensional shape of large spherical viruses. The faceted shape depends only on the dimensionless Foppl-von Karman number \gamma=YR^2/\kappa, where Y is the two-dimensional Young's modulus of the protein shell, \kappa is its bending rigidity and R is the mean virus radius. The shape can be parameterized more quantitatively in terms of a spherical harmonic expansion. We also investigate elastic shell theory for extremely large \gamma, 10^3 < \gamma < 10^8, and find results applicable to icosahedral shapes of large vesicles studied with freeze fracture and electron microscopy.Comment: 11 pages, 12 figure

A Unified Algebraic Approach to Few and Many-Body Correlated Systems

The present article is an extended version of the paper {\it Phys. Rev.} {\bf B 59}, R2490 (1999), where, we have established the equivalence of the Calogero-Sutherland model to decoupled oscillators. Here, we first employ the same approach for finding the eigenstates of a large class of Hamiltonians, dealing with correlated systems. A number of few and many-body interacting models are studied and the relationship between their respective Hilbert spaces, with that of oscillators, is found. This connection is then used to obtain the spectrum generating algebras for these systems and make an algebraic statement about correlated systems. The procedure to generate new solvable interacting models is outlined. We then point out the inadequacies of the present technique and make use of a novel method for solving linear differential equations to diagonalize the Sutherland model and establish a precise connection between this correlated system's wave functions, with those of the free particles on a circle. In the process, we obtain a new expression for the Jack polynomials. In two dimensions, we analyze the Hamiltonian having Laughlin wave function as the ground-state and point out the natural emergence of the underlying linear $W_{1+\infty}$ symmetry in this approach.Comment: 18 pages, Revtex format, To appear in Physical Review

Equivalence of the Calogero-Sutherland Model to Free Harmonic Oscillators

A similarity transformation is constructed through which a system of particles interacting with inverse-square two-body and harmonic potentials in one dimension, can be mapped identically, to a set of free harmonic oscillators. This equivalence provides a straightforward method to find the complete set of eigenfunctions, the exact constants of motion and a linear $W_{1+\infty}$ algebra associated with this model. It is also demonstrated that a large class of models with long-range interactions, both in one and higher dimensions can be made equivalent to decoupled oscillators.Comment: 9 pages, REVTeX, Completely revised, few new equations and references are adde

Magnetic ground state of the two isostructual polymeric quantum magnets [Cu(HF2)(pyrazine)2]SbF6 and [Co(HF2)(pyrazine)2]SbF6 investigated with neutron powder diffraction

The magnetic ground state of two isostructural coordination polymers, (i) the quasi-two-dimensional S=1/2 square-lattice antiferromagnet [Cu(HF2)(pyrazine)2]SbF6 and (ii) a related compound [Co(HF2)(pyrazine)2]SbF6, was examined with neutron powder diffraction measurements. We find that the ordered moments of the Heisenberg S=1/2 Cu(II) ions in [Cu(HF2)(pyrazine)2]SbF6 are 0.6(1)μb, while the ordered moments for the Co(II) ions in [Co(HF2)(pyrazine)2]SbF6 are 3.02(6)μb. For Cu(II), this reduced moment indicates the presence of quantum fluctuations below the ordering temperature. We show from heat capacity and electron spin resonance measurements that due to the crystal electric field splitting of the S=3/2 Co(II) ions in [Co(HF2)(pyrazine)2]SbF6, this isostructual polymer also behaves as an effective spin-half magnet at low temperatures. The Co moments in [Co(HF2)(pyrazine)2]SbF6 show strong easy-axis anisotropy, neutron diffraction data, which do not support the presence of quantum fluctuations in the ground state, and heat capacity data, which are consistent with 2D or close to 3D spatial exchange anisotropy

Water as a catalytic switch in the oxidation of aryl alcohols by polymer incarcerated rhodium nanoparticles

One of the major goals in the oxidation of organic substrates, and especially for alcohol oxidation, is the use of molecular oxygen as the oxidant under mild conditions. Here we report the synthesis and testing of Rh polymer incarcerated catalysts, using a metal so far not used for alcohol oxidation reactions, in which the catalytic activity towards aryl alcohol oxidation, for substrates like 1-phenylethanol and benzyl alcohol, is switched on by the addition of water as co-solvent in toluene. This is done by using air as oxidant at atmospheric pressure, in one of the mildest reaction conditions reported for this class of reaction. The promoting effect of water to higher conversions was observed also for rhodium over alumina supported catalysts, which were used as a benchmark allowing in all cases high conversion and selectivity to the ketone or the aldehyde within a short reaction time. The effect of water was explained as a medium capable to promote the oxidation of the alcohol to the ketone in a biphasic system assisted by phase transfer catalysis. This is particularly relevant for alcohols like 1-phenylethanol or benzyl alcohol that are not soluble in water at room temperature, and for which alternative oxidation routes are needed, as well as to switch on the catalytic activity of metal nanoparticles in a facile and green manner for the activation of molecular oxygen. Aliphatic alcohols like 1-octanol and 3-octanol were also tested, still showing Rh based catalysts as promising materials for this reaction if toluene only was used as solvent instead