Dynamics of morphogen source formation in a growing tissue.

A tight regulation of morphogen production is key for morphogen gradient formation and thereby for reproducible and organised organ development. Although many genetic interactions involved in the establishment of morphogen production domains are known, the biophysical mechanisms of morphogen source formation are poorly understood. Here we addressed this by focusing on the morphogen Sonic hedgehog (Shh) in the vertebrate neural tube. Shh is produced by the adjacently located notochord and by the floor plate of the neural tube. Using a data-constrained computational screen, we identified different possible mechanisms by which floor plate formation can occur, only one of which is consistent with experimental data. In this mechanism, the floor plate is established rapidly in response to Shh from the notochord and the dynamics of regulatory interactions within the neural tube. In this process, uniform activators and Shh-dependent repressors are key for establishing the floor plate size. Subsequently, the floor plate becomes insensitive to Shh and increases in size due to tissue growth, leading to scaling of the floor plate with neural tube size. In turn, this results in scaling of the Shh amplitude with tissue growth. Thus, this mechanism ensures a separation of time scales in floor plate formation, so that the floor plate domain becomes growth-dependent after an initial rapid establishment phase. Our study raises the possibility that the time scale separation between specification and growth might be a common strategy for scaling the morphogen gradient amplitude in growing organs. The model that we developed provides a new opportunity for quantitative studies of morphogen source formation in growing tissues.

Chaotic Properties of a Turbulent Isotropic Fluid.

By tracking the divergence of two initially close trajectories in phase space in an Eulerian approach to forced turbulence, the relation between the maximal Lyapunov exponent λ and the Reynolds number Re is measured using direct numerical simulations, performed on up to 2048^{3} collocation points. The Lyapunov exponent is found to solely depend on the Reynolds number with λ∝Re^{0.53} and that after a transient period the divergence of trajectories grows at the same rate at all scales. Finally a linear divergence is seen that is dependent on the energy forcing rate. Links are made with other chaotic systems.

Fully resolved array of simulations investigating the influence of the magnetic Prandtl number on magnetohydrodynamic turbulence.

We explore the effect of the magnetic Prandtl number Pm on energy and dissipation in fully resolved direct numerical simulations of steady-state, mechanically forced, homogeneous magnetohydrodynamic turbulence in the range 1/32