Transport in chaotic systems

This dissertation addresses the general problem of transport in chaotic systems. Typical fluid problem of the kind is the advection and diffusion of a passive scalar. The magnetic field evolution in a chaotic conducting media is an example of the chaotic transport of a vector field. In kinetic theory, the collisional relaxation of a distribution function in phase space is also an advection-diffusion problem, but in a higher dimensional space.;In a chaotic flow neighboring points tend to separate exponentially in time, exp({dollar}\omega t{dollar}) with {dollar}\omega{dollar} the Liapunov exponent. The characteristic parameter for the transport of a scalar in a chaotic flow is {dollar}\Omega\ \equiv\ \omega L\sp2/D{dollar} where L is the spatial scale and D is the diffusivity. For {dollar}\Omega\ \gg\ 1{dollar}, the scalar is advected with the flow for a time {dollar}t\sb{lcub}a{rcub}\ \equiv{dollar} ln(2{dollar}\Omega{dollar})/2{dollar}\omega{dollar} and then diffuses during the relatively short period 1/{dollar}\omega{dollar} centered on the time {dollar}t\sb{lcub}a{rcub}{dollar}. This rapid diffusion occurs only along the field line of the {dollar}\rm \ s\sb\infty{dollar} vector, which defines the stable direction for neighboring streamlines to converge. Diffusion is impeded at the sharp bends of an {dollar}\rm \ s{dollar} line because of a peculiarly small finite time Lyapunov exponent, hence a class of diffusion barriers is created inside a chaotic sea. This result comes from a fundamental relationship between the finite time Lyapunov exponent and the geometry of the {dollar}\rm \ s{dollar} lines, which we rigorously show in 2D and numerically validated for 3D flows.;The evolution of a general 3D magnetic field in a highly conducting chaotic media is also related to the spatial-temporal dependence of the finite time Lyapunov exponent. The Ohmic dissipation in a chaotic plasma will become a dominate process despite a small plasma resistivity. We show that the Ohmic heating in a chaotic plasma occurs in current filaments or current sheets. The particular form is determined by the time dependence of spatial gradient of the finite time Lyapunov exponent along a direction in which neighboring point neither diverge nor converge

Elevated expression of human bHLH factor ATOH7 accelerates cell cycle progression of progenitors and enhances production of avian retinal ganglion cells.

The production of vertebrate retinal projection neurons, retinal ganglion cells (RGCs), is regulated by cell-intrinsic determinants and cell-to-cell signaling events. The basic-helix-loop-helix (bHLH) protein Atoh7 is a key neurogenic transcription factor required for RGC development. Here, we investigate whether manipulating human ATOH7 expression among uncommitted progenitors can promote RGC fate specification and thus be used as a strategy to enhance RGC genesis. Using the chicken retina as a model, we show that cell autonomous expression of ATOH7 is sufficient to induce precocious RGC formation and expansion of the neurogenic territory. ATOH7 overexpression among neurogenic progenitors significantly enhances RGC production at the expense of reducing the progenitor pool. Furthermore, forced expression of ATOH7 leads to a minor increase of cone photoreceptors. We provide evidence that elevating ATOH7 levels accelerates cell cycle progression from S to M phase and promotes cell cycle exit. We also show that ATOH7-induced ectopic RGCs often exhibit aberrant axonal projection patterns and are correlated with increased cell death during the period of retinotectal connections. These results demonstrate the high potency of human ATOH7 in promoting early retinogenesis and specifying the RGC differentiation program, thus providing insight for manipulating RGC production from stem cell-derived retinal organoids